Class 7
Mathematics
Federal Board of Intermediate and Secondary Education (FBISE) (FBISE)
📚 Model Textbook of Mathematics Grade 7 (FBISE / NBF)
Class 7 Mathematics Ch 6 Mastery Guide: Algebraic Expressions, Polynomials, Standard Identities, Factorization & Sequences (FBISE)
📖 Chapter 6: Algebra
📅 Updated: Sep 11, 2026
Teacher & Parent Roadmap
Grade 7 Mathematics • FBISE / National Curriculum (NBF)
Instructional Guide: Unit 06 Algebra
Target Learning Outcomes (SLOs)
- Define variables, constants, coefficients, and build algebraic expressions from word problems.
- Define a polynomial as an algebraic expression with non-negative whole-number exponents ($0, 1, 2, \dots$); distinguish expressions that are NOT polynomials.
- Determine the degree of a polynomial and write polynomials in standard descending order with their leading coefficient.
- Classify polynomials by terms: monomial (1 term), binomial (2 terms), trinomial (3 terms).
- Perform addition (combining like terms) and subtraction (distributing the negative sign) using horizontal and vertical methods.
- Multiply polynomials: monomial by polynomial, binomial by binomial, and binomial by trinomial using distributive law and grid area models.
- Prove and apply 4 fundamental algebraic identities:
- $(a + b)^2 = a^2 + 2ab + b^2$
- $(a - b)^2 = a^2 - 2ab + b^2$
- $(x + a)(x + b) = x^2 + (a + b)x + ab$
- $(a + b)(a - b) = a^2 - b^2$
- Perform rapid mental arithmetic using algebraic identities (e.g. $102^2, 1.98^2, 52 \times 48, 103 \times 102$).
- Factorize algebraic expressions using common monomial factoring, grouping 4 terms, and splitting the middle term for quadratic trinomials $ax^2 + bx + c$.
- Identify and model number patterns and sequences (Fibonacci numbers in nature, arithmetic vs geometric sequences, general term $T_n$, stick patterns).
Prerequisites & Bridge Concepts
- Signed Numbers: $(+) \times (+) = (+)$, $(-) \times (-) = (+)$, $(+) \times (-) = (-)$. For addition: same signs add and keep sign; different signs subtract and take the sign of the larger magnitude.
- Laws of Exponents: $x^m \times x^n = x^{m+n}$. When multiplying like variables, add their exponents ($x^2 \cdot x^3 = x^5$).
- Distributive Property: $a(b + c) = ab + ac$. The factor outside the bracket multiplies every term inside.
- Geometric Mensuration: Perimeter of rectangle $= 2(L + W)$, Area of rectangle $= L \times W$, Perimeter of square $= 4s$, Area of square $= s^2$.
Common Misconceptions & Pitfalls
- The Binomial Squaring Trap: Students often mistakenly write $(a + b)^2 = a^2 + b^2$. False! Never forget the cross term $+2ab$. Geometrically, $(a+b)^2$ contains two extra rectangles of area $ab$.
- Denominator Variable in Polynomials: Expressions like $x + \frac{1}{x}$ or $\frac{7}{y}$ are NOT polynomials because $\frac{1}{x} = x^{-1}$, which has a negative power. Exponents must be non-negative whole numbers $\{0, 1, 2, \dots\}$.
- Sign Inversion in Subtraction: When subtracting $(4x - 10)$ from $(8x - 12)$, the negative sign must be distributed to BOTH terms: $-(4x - 10) = -4x + 10$.
- "Subtract $A$ from $B$": This means $B - A$, NOT $A - B$. The expression following "from" always comes first!
- Middle Term Factoring Signs: In $x^2 - 5x - 6$, the pair of numbers multiplying to $-6$ and adding to $-5$ is $-6$ and $+1$, NOT $-2$ and $-3$ (which multiply to $+6$). Check both sum and product!
Pedagogical Tips & Real-World Connections
Introduce algebra as generalized arithmetic—a language where letters stand for numbers so formulas work universally. Connect identities to geometric area models (cutting out cardboard squares and rectangles). Connect sequences to nature: show sunflower seed heads or pine cones where seeds spiral in consecutive Fibonacci numbers ($1, 1, 2, 3, 5, 8, 13, 21, 34, \dots$).
Quick Jump Navigation (Unit 06 Algebra)
The Birth of Algebra: Al-Khwarizmi's Mathematical Revolution
Father of Algebra: Muhammad ibn Musa al-Khwarizmi (c. 780–850 CE)
Muhammad ibn Musa al-Khwarizmi was a 9th-century Muslim mathematician, astronomer, and scholar at the famous House of Wisdom (*Bayt al-Hikmah*) in Baghdad. He is universally celebrated as the Father of Algebra. The very word "Algebra" comes from the Arabic word al-jabr in the title of his groundbreaking book, Kitab al-Jabr wa'l-Muqabala ("The Compendious Book on Calculation by Completion and Balancing").
Al-Khwarizmi introduced the revolutionary method of expressing unknown quantities as symbols and manipulating them with systematic rules. In fact, our modern mathematical term "Algorithm" is directly derived from his Latinized name, Algoritmi!
Unit 06 Algebra: Complete Conceptual Hierarchy
1. Foundations
Variables & Constants
Algebraic Expressions
Monomial, Binomial, Trinomial
Degree & Leading Coeff
Variables & Constants
Algebraic Expressions
Monomial, Binomial, Trinomial
Degree & Leading Coeff
2. Operations
Addition (Like terms)
Subtraction (Sign flip)
Horizontal & Vertical
Multiplication (Grid/Distr.)
Addition (Like terms)
Subtraction (Sign flip)
Horizontal & Vertical
Multiplication (Grid/Distr.)
3. 4 Standard Identities
$(a+b)^2 = a^2+2ab+b^2$
$(a-b)^2 = a^2-2ab+b^2$
$(x+a)(x+b)$
$(a+b)(a-b) = a^2-b^2$
$(a+b)^2 = a^2+2ab+b^2$
$(a-b)^2 = a^2-2ab+b^2$
$(x+a)(x+b)$
$(a+b)(a-b) = a^2-b^2$
4. Factorization
Common Monomial Factor
Grouping 4 Terms
Splitting Middle Term
Trinomial $ax^2+bx+c$
Common Monomial Factor
Grouping 4 Terms
Splitting Middle Term
Trinomial $ax^2+bx+c$
5. Sequences & Patterns
Fibonacci in Nature
Arithmetic Sequences ($d$)
Geometric Sequences ($r$)
General Term $T_n$
Fibonacci in Nature
Arithmetic Sequences ($d$)
Geometric Sequences ($r$)
General Term $T_n$
1. Algebraic Expressions, Polynomials & Operations
Core Theory: What is an Algebraic Expression?
Algebra is generalized arithmetic. While arithmetic deals with specific known numbers (like $3 + 5 = 8$), algebra uses letters called variables to represent unknown or changing quantities.
- Variable: A letter or symbol (such as $x, y, z$) representing an unknown quantity. E.g., if Qasim's age is unknown, we call it $y$. His brother Furqan is 3 years older, so his age is $y + 3$.
- Constant: A quantity with a fixed numerical value that never changes (such as $2, 6, 0, -3, 12$).
- Coefficient: The numerical multiplying factor appearing directly before a variable. In $9y$, the coefficient is $9$. In $x$, the coefficient is $1$. In $-5z$, the coefficient is $-5$.
- Terms: The individual parts of an algebraic expression separated by $+$ or $-$ signs. E.g., in $5x - 3$, there are two terms: $5x$ and $-3$. In $x^2 - 3xz + yz$, there are three terms: $x^2$, $-3xz$, and $yz$.
Definition of a Polynomial:
A polynomial is an algebraic expression consisting of variables and coefficients, involving only addition, subtraction, multiplication, and non-negative integer (whole number) exponents ($0, 1, 2, 3, \dots$).
Warning: An expression is NOT a polynomial if any variable has a negative exponent, a fractional exponent, or appears in the denominator (e.g., $\\frac{1}{x} = x^{-1}$ or $\\frac{7}{y} = 7y^{-1}$).
Monomial (1 Term)
$12$ (degree 0)
$4x$ (degree 1)
$7y^2$ (degree 2)
$-15z^6$ (degree 6)
$12$ (degree 0)
$4x$ (degree 1)
$7y^2$ (degree 2)
$-15z^6$ (degree 6)
Binomial (2 Terms)
$2x - 13$
$7x^2y^3 + 4x^2z$
$x^4 - 3$
$6x + 7xy$
$2x - 13$
$7x^2y^3 + 4x^2z$
$x^4 - 3$
$6x + 7xy$
Trinomial (3 Terms)
$x + 4y - 8$
$5x^2 + x - 6$
$p^2 + pq + 8$
$6x^2 + 3x + 2$
$x + 4y - 8$
$5x^2 + x - 6$
$p^2 + pq + 8$
$6x^2 + 3x + 2$
Degree & Leading Coeff
Degree: Greatest exponent sum in any single term.
Leading Coeff: Coefficient of the highest degree term when arranged in descending order.
Degree: Greatest exponent sum in any single term.
Leading Coeff: Coefficient of the highest degree term when arranged in descending order.
Exercise 6.1 • 100% Complete Step-by-Step Solved Solutions
Question 1
Exercise 6.1
Indicate whether each of the following expressions are polynomials or not. (b) If it is a polynomial, find its degree and classify it by the number of terms. (c) Identify the leading coefficient in the case of the polynomial:
(i) $6x + 4$: Exponent of $x$ is $1$ (a whole number). It is a polynomial.
• Degree: $1$ • Terms: 2 (Binomial) • Leading coefficient: $6$.
• Degree: $1$ • Terms: 2 (Binomial) • Leading coefficient: $6$.
(ii) $x^2 - 2x + 1$: All exponents ($2, 1, 0$) are whole numbers. It is a polynomial.
• Degree: $2$ • Terms: 3 (Trinomial) • Leading coefficient: $1$.
• Degree: $2$ • Terms: 3 (Trinomial) • Leading coefficient: $1$.
(iii) $y^2 - 2 + \frac{1}{y^2}$: The term $\frac{1}{y^2} = y^{-2}$ has a negative exponent $-2$.
• Not a polynomial: A polynomial cannot have a variable in the denominator.
• Not a polynomial: A polynomial cannot have a variable in the denominator.
(iv) $x - 3 + \frac{1}{x}$: The term $\frac{1}{x} = x^{-1}$ has a negative exponent $-1$.
• Not a polynomial: A polynomial cannot have a variable in the denominator.
• Not a polynomial: A polynomial cannot have a variable in the denominator.
(v) $\frac{5}{2}x^2 - 2x^4$: In standard descending form: $-2x^4 + \frac{5}{2}x^2$. Exponents $4$ and $2$ are whole numbers. It is a polynomial.
• Degree: $4$ • Terms: 2 (Binomial) • Leading coefficient: $-2$.
• Degree: $4$ • Terms: 2 (Binomial) • Leading coefficient: $-2$.
(vi) $6 + \frac{7}{y}$: The term $\frac{7}{y} = 7y^{-1}$ contains a negative power.
• Not a polynomial: A polynomial cannot have a variable in the denominator.
• Not a polynomial: A polynomial cannot have a variable in the denominator.
(vii) $y + 6y^3$: In standard descending form: $6y^3 + y$. All exponents are whole numbers. It is a polynomial.
• Degree: $3$ • Terms: 2 (Binomial) • Leading coefficient: $6$.
• Degree: $3$ • Terms: 2 (Binomial) • Leading coefficient: $6$.
(viii) $x + \frac{1}{x}$: Contains $\frac{1}{x} = x^{-1}$.
• Not a polynomial: A polynomial cannot have a variable in the denominator.
• Not a polynomial: A polynomial cannot have a variable in the denominator.
(ix) $\frac{1}{x^2} + \frac{1}{x^3}$: Both terms have negative powers ($x^{-2} + x^{-3}$).
• Not a polynomial: Variables cannot appear in the denominator.
• Not a polynomial: Variables cannot appear in the denominator.
(x) $\frac{3}{7} - 4x + 6x^3 - 3x^2$: Arranged in descending order: $6x^3 - 3x^2 - 4x + \frac{3}{7}$. All powers are whole numbers ($3, 2, 1, 0$). It is a polynomial.
• Degree: $3$ • Terms: 4 (Polynomial of 4 terms) • Leading coefficient: $6$.
• Degree: $3$ • Terms: 4 (Polynomial of 4 terms) • Leading coefficient: $6$.
Question 2
Exercise 6.1
Categorize the following polynomials as monomial, binomial or trinomial:
(i) $7x$: Exactly 1 term → Monomial
(ii) $6x^2 + 3x + 2$: Exactly 3 terms → Trinomial
(iii) $6x^2 + x - 10x^3$: Exactly 3 terms → Trinomial
(iv) $-10$: Single constant term → Monomial
(v) $x^3 - 10xy^2$: Exactly 2 terms → Binomial
(vi) $6x + 7xy$: Exactly 2 terms → Binomial
(vii) $y^2 - 6x^2y^2 + y^3$: Exactly 3 terms → Trinomial
(viii) $x^2yz + 5y^2z + 6x^2z^2$: Exactly 3 terms → Trinomial
(ix) $x^4 - 3$: Exactly 2 terms → Binomial
(x) $3 - x^3 + 10x$: Exactly 3 terms → Trinomial
Question 3
Exercise 6.1
Find the sum of the following polynomials:
(i) $7x + 4$ and $9x - 3$:
Sum $= (7x + 4) + (9x - 3) = (7 + 9)x + (4 - 3) = \mathbf{16x + 1}$.
Sum $= (7x + 4) + (9x - 3) = (7 + 9)x + (4 - 3) = \mathbf{16x + 1}$.
(ii) $a^2 - 8$ and $a^2 + 9$:
Sum $= (a^2 - 8) + (a^2 + 9) = (1 + 1)a^2 + (-8 + 9) = \mathbf{2a^2 + 1}$.
Sum $= (a^2 - 8) + (a^2 + 9) = (1 + 1)a^2 + (-8 + 9) = \mathbf{2a^2 + 1}$.
(iii) $9x^2 + x - 3$ and $6x^2 - 3x + 4$:
Sum $= (9x^2 + 6x^2) + (x - 3x) + (-3 + 4) = \mathbf{15x^2 - 2x + 1}$.
Sum $= (9x^2 + 6x^2) + (x - 3x) + (-3 + 4) = \mathbf{15x^2 - 2x + 1}$.
(iv) $a^2 + 2ab + b^2$ and $a^2 - 2ab + b^2$:
Sum $= (a^2 + a^2) + (2ab - 2ab) + (b^2 + b^2) = 2a^2 + 0 + 2b^2 = \mathbf{2a^2 + 2b^2}$.
Sum $= (a^2 + a^2) + (2ab - 2ab) + (b^2 + b^2) = 2a^2 + 0 + 2b^2 = \mathbf{2a^2 + 2b^2}$.
(v) $x^3 + 5x^2 - 6x + 7$ and $2x^3 + 7x^2 - 10x + 7$:
Sum $= (1 + 2)x^3 + (5 + 7)x^2 + (-6 - 10)x + (7 + 7) = \mathbf{3x^3 + 12x^2 - 16x + 14}$.
Sum $= (1 + 2)x^3 + (5 + 7)x^2 + (-6 - 10)x + (7 + 7) = \mathbf{3x^3 + 12x^2 - 16x + 14}$.
(vi) $2a + 9b + c - 4d$ and $-a - 7b - 2c + 5d$:
Sum $= (2 - 1)a + (9 - 7)b + (1 - 2)c + (-4 + 5)d = \mathbf{a + 2b - c + d}$.
Sum $= (2 - 1)a + (9 - 7)b + (1 - 2)c + (-4 + 5)d = \mathbf{a + 2b - c + d}$.
(vii) $4a^2 - 7b - 4b^2$ and $2a^2 + 10b + 3b^2$:
Sum $= (4 + 2)a^2 + (-7 + 10)b + (-4 + 3)b^2 = \mathbf{6a^2 + 3b - b^2}$.
Sum $= (4 + 2)a^2 + (-7 + 10)b + (-4 + 3)b^2 = \mathbf{6a^2 + 3b - b^2}$.
(viii) $3x^2 + x - 2$, $2x^2 - 2x + 3$, and $x^2 + 4x + 2$:
Sum $= (3 + 2 + 1)x^2 + (1 - 2 + 4)x + (-2 + 3 + 2) = \mathbf{6x^2 + 3x + 3}$.
Sum $= (3 + 2 + 1)x^2 + (1 - 2 + 4)x + (-2 + 3 + 2) = \mathbf{6x^2 + 3x + 3}$.
(ix) $s^2 + 3t^2 + 4st$, $2s^2 + 4t^2 - 3st$, and $4s^2 - 2t^2 + 9st$:
Sum $= (1 + 2 + 4)s^2 + (3 + 4 - 2)t^2 + (4 - 3 + 9)st = \mathbf{7s^2 + 5t^2 + 10st}$.
Sum $= (1 + 2 + 4)s^2 + (3 + 4 - 2)t^2 + (4 - 3 + 9)st = \mathbf{7s^2 + 5t^2 + 10st}$.
Question 4
Exercise 6.1
Find the perimeter of each rectangle:
(i) Length $= 11x$, Width $= 8x - 10$:
Perimeter $= 2(\text{Length} + \text{Width}) = 2(11x + 8x - 10)$
$= 2(19x - 10) = \mathbf{38x - 20}$.
Perimeter $= 2(\text{Length} + \text{Width}) = 2(11x + 8x - 10)$
$= 2(19x - 10) = \mathbf{38x - 20}$.
(ii) Length $= 3x + 6$, Width $= 3x$:
Perimeter $= 2(3x + 6 + 3x) = 2(6x + 6)$
$= 2 \times 6(x + 1) = \mathbf{12x + 12}$.
Perimeter $= 2(3x + 6 + 3x) = 2(6x + 6)$
$= 2 \times 6(x + 1) = \mathbf{12x + 12}$.
(iii) Length $= 4x - 2$, Width $= 2x + 7$:
Perimeter $= 2(4x - 2 + 2x + 7) = 2(6x + 5)$
$= \mathbf{12x + 10}$.
Perimeter $= 2(4x - 2 + 2x + 7) = 2(6x + 5)$
$= \mathbf{12x + 10}$.
Question 5
Exercise 6.1
Subtract:
Rule: To subtract $A$ from $B$, we compute $B - A$. Invert the sign of every term inside $A$ and combine like terms.
(i) $a + b + c$ from $2a + 2b - c$:
$= (2a + 2b - c) - (a + b + c) = 2a + 2b - c - a - b - c$
$= (2 - 1)a + (2 - 1)b + (-1 - 1)c = \mathbf{a + b - 2c}$.
$= (2a + 2b - c) - (a + b + c) = 2a + 2b - c - a - b - c$
$= (2 - 1)a + (2 - 1)b + (-1 - 1)c = \mathbf{a + b - 2c}$.
(ii) $5x^2 + x - 9$ from $7x^2 - 2x + 10$:
$= (7x^2 - 2x + 10) - (5x^2 + x - 9) = 7x^2 - 2x + 10 - 5x^2 - x + 9$
$= (7 - 5)x^2 + (-2 - 1)x + (10 + 9) = \mathbf{2x^2 - 3x + 19}$.
$= (7x^2 - 2x + 10) - (5x^2 + x - 9) = 7x^2 - 2x + 10 - 5x^2 - x + 9$
$= (7 - 5)x^2 + (-2 - 1)x + (10 + 9) = \mathbf{2x^2 - 3x + 19}$.
(iii) $6a^2 - 10ab - b^2$ from $-2a^2 + 5ab - 3b^2$:
$= (-2a^2 + 5ab - 3b^2) - (6a^2 - 10ab - b^2) = -2a^2 + 5ab - 3b^2 - 6a^2 + 10ab + b^2$
$= (-2 - 6)a^2 + (5 + 10)ab + (-3 + 1)b^2 = \mathbf{-8a^2 + 15ab - 2b^2}$.
$= (-2a^2 + 5ab - 3b^2) - (6a^2 - 10ab - b^2) = -2a^2 + 5ab - 3b^2 - 6a^2 + 10ab + b^2$
$= (-2 - 6)a^2 + (5 + 10)ab + (-3 + 1)b^2 = \mathbf{-8a^2 + 15ab - 2b^2}$.
(iv) $4a^2 + 3b^2 - 4ab$ from $4a^2 + 3b^2 - 6ab$:
$= (4a^2 + 3b^2 - 6ab) - (4a^2 + 3b^2 - 4ab) = 4a^2 + 3b^2 - 6ab - 4a^2 - 3b^2 + 4ab$
$= (4 - 4)a^2 + (3 - 3)b^2 + (-6 + 4)ab = 0 + 0 - 2ab = \mathbf{-2ab}$.
$= (4a^2 + 3b^2 - 6ab) - (4a^2 + 3b^2 - 4ab) = 4a^2 + 3b^2 - 6ab - 4a^2 - 3b^2 + 4ab$
$= (4 - 4)a^2 + (3 - 3)b^2 + (-6 + 4)ab = 0 + 0 - 2ab = \mathbf{-2ab}$.
(v) $10x^3 - 8x^2 + 4x + 3$ from $8x^3 + 4x^2 - x + 5$:
$= (8x^3 + 4x^2 - x + 5) - (10x^3 - 8x^2 + 4x + 3)$
$= 8x^3 + 4x^2 - x + 5 - 10x^3 + 8x^2 - 4x - 3$
$= (8 - 10)x^3 + (4 + 8)x^2 + (-1 - 4)x + (5 - 3) = \mathbf{-2x^3 + 12x^2 - 5x + 2}$.
$= (8x^3 + 4x^2 - x + 5) - (10x^3 - 8x^2 + 4x + 3)$
$= 8x^3 + 4x^2 - x + 5 - 10x^3 + 8x^2 - 4x - 3$
$= (8 - 10)x^3 + (4 + 8)x^2 + (-1 - 4)x + (5 - 3) = \mathbf{-2x^3 + 12x^2 - 5x + 2}$.
(vi) $x^2 + y^2 - xy$ from $3x^2 - 2y^2 + 4xy$:
$= (3x^2 - 2y^2 + 4xy) - (x^2 + y^2 - xy) = 3x^2 - 2y^2 + 4xy - x^2 - y^2 + xy$
$= (3 - 1)x^2 + (-2 - 1)y^2 + (4 + 1)xy = \mathbf{2x^2 - 3y^2 + 5xy}$.
$= (3x^2 - 2y^2 + 4xy) - (x^2 + y^2 - xy) = 3x^2 - 2y^2 + 4xy - x^2 - y^2 + xy$
$= (3 - 1)x^2 + (-2 - 1)y^2 + (4 + 1)xy = \mathbf{2x^2 - 3y^2 + 5xy}$.
2. Multiplication of Polynomials
Theory: Laws & Visual Area Model of Multiplication
In addition and subtraction, only like terms can be combined. However, in multiplication, any two terms can be multiplied together—even if they have completely different variables!
Fundamental Multiplication Rules:
- Signs: $(+x)(+y) = +xy$, $(-x)(-y) = +xy$, and $(+x)(-y) = -xy$. Like signs yield positive; unlike signs yield negative.
- Product of Powers: $x^m \times x^n = x^{m+n}$. Coefficients multiply normally, while exponents of identical variables add together: $(8x^2 y^2 z)(2xyz) = (8 \times 2)x^{2+1}y^{2+1}z^{1+1} = 16x^3 y^3 z^2$.
- Distributive Law: $a(b + c + d) = ab + ac + ad$.
- Binomial Product: $(a + b)(c + d) = a(c + d) + b(c + d) = ac + ad + bc + bd$.
The Grid / Area Model Method:
To visualize $(x - 4)(3x + 2)$, draw a 2x2 grid representing a rectangle broken into 4 smaller areas:
| $\times$ | $3x$ | $+2$ |
| $x$ | $3x^2$ | $+2x$ |
| $-4$ | $-12x$ | $-8$ |
Sum of all 4 cells: $3x^2 + 2x - 12x - 8 = \mathbf{3x^2 - 10x - 8}$.
Exercise 6.2 • 100% Complete Step-by-Step Solved Solutions
Question 1
Exercise 6.2
Find the product of the following:
(i) $3x$ and $9x^2$:
Product $= (3 \times 9) \cdot (x^{1+2}) = \mathbf{27x^3}$.
Product $= (3 \times 9) \cdot (x^{1+2}) = \mathbf{27x^3}$.
(ii) $6x$ and $4xy$:
Product $= (6 \times 4) \cdot (x^{1+1} y) = \mathbf{24x^2y}$.
Product $= (6 \times 4) \cdot (x^{1+1} y) = \mathbf{24x^2y}$.
(iii) $7x^2$ and $(7x + 3)$:
Product $= 7x^2(7x) + 7x^2(3) = (7 \times 7)x^{2+1} + (7 \times 3)x^2 = \mathbf{49x^3 + 21x^2}$.
Product $= 7x^2(7x) + 7x^2(3) = (7 \times 7)x^{2+1} + (7 \times 3)x^2 = \mathbf{49x^3 + 21x^2}$.
(iv) $-3x$ and $(6x^2 + 4x)$:
Product $= (-3x)(6x^2) + (-3x)(4x) = (-3 \times 6)x^{1+2} + (-3 \times 4)x^{1+1} = \mathbf{-18x^3 - 12x^2}$.
Product $= (-3x)(6x^2) + (-3x)(4x) = (-3 \times 6)x^{1+2} + (-3 \times 4)x^{1+1} = \mathbf{-18x^3 - 12x^2}$.
(v) $4x$ and $(3x^2 + 7x - 6)$:
Product $= (4x)(3x^2) + (4x)(7x) - (4x)(6) = (4 \times 3)x^{1+2} + (4 \times 7)x^{1+1} - 24x = \mathbf{12x^3 + 28x^2 - 24x}$.
Product $= (4x)(3x^2) + (4x)(7x) - (4x)(6) = (4 \times 3)x^{1+2} + (4 \times 7)x^{1+1} - 24x = \mathbf{12x^3 + 28x^2 - 24x}$.
Question 2
Exercise 6.2
Evaluate the following:
(i) $(2x - 3)(3x - 4)$:
$= 2x(3x - 4) - 3(3x - 4) = 6x^2 - 8x - 9x + 12 = \mathbf{6x^2 - 17x + 12}$.
$= 2x(3x - 4) - 3(3x - 4) = 6x^2 - 8x - 9x + 12 = \mathbf{6x^2 - 17x + 12}$.
(ii) $(2x - 7)(4x + 8)$:
$= 2x(4x + 8) - 7(4x + 8) = 8x^2 + 16x - 28x - 56 = \mathbf{8x^2 - 12x - 56}$.
$= 2x(4x + 8) - 7(4x + 8) = 8x^2 + 16x - 28x - 56 = \mathbf{8x^2 - 12x - 56}$.
(iii) $(6a - 2)(6a^2 + a - 5)$:
$= 6a(6a^2 + a - 5) - 2(6a^2 + a - 5)$
$= 36a^3 + 6a^2 - 30a - 12a^2 - 2a + 10$
$= 36a^3 + (6 - 12)a^2 + (-30 - 2)a + 10 = \mathbf{36a^3 - 6a^2 - 32a + 10}$.
$= 6a(6a^2 + a - 5) - 2(6a^2 + a - 5)$
$= 36a^3 + 6a^2 - 30a - 12a^2 - 2a + 10$
$= 36a^3 + (6 - 12)a^2 + (-30 - 2)a + 10 = \mathbf{36a^3 - 6a^2 - 32a + 10}$.
(iv) $(4a^2 - 3a + 6)(2a + 5)$:
$= 2a(4a^2 - 3a + 6) + 5(4a^2 - 3a + 6)$
$= 8a^3 - 6a^2 + 12a + 20a^2 - 15a + 30$
$= 8a^3 + (-6 + 20)a^2 + (12 - 15)a + 30 = \mathbf{8a^3 + 14a^2 - 3a + 30}$.
$= 2a(4a^2 - 3a + 6) + 5(4a^2 - 3a + 6)$
$= 8a^3 - 6a^2 + 12a + 20a^2 - 15a + 30$
$= 8a^3 + (-6 + 20)a^2 + (12 - 15)a + 30 = \mathbf{8a^3 + 14a^2 - 3a + 30}$.
(v) $(x + y)(x^2 - xy)$:
$= x(x^2 - xy) + y(x^2 - xy) = x^3 - x^2y + x^2y - xy^2 = \mathbf{x^3 - xy^2}$.
$= x(x^2 - xy) + y(x^2 - xy) = x^3 - x^2y + x^2y - xy^2 = \mathbf{x^3 - xy^2}$.
(vi) $(x - y)(x^2 + xy + y^2)$:
$= x(x^2 + xy + y^2) - y(x^2 + xy + y^2)$
$= x^3 + x^2y + xy^2 - x^2y - xy^2 - y^3 = \mathbf{x^3 - y^3}$ (Formula for difference of cubes).
$= x(x^2 + xy + y^2) - y(x^2 + xy + y^2)$
$= x^3 + x^2y + xy^2 - x^2y - xy^2 - y^3 = \mathbf{x^3 - y^3}$ (Formula for difference of cubes).
(vii) $(4x^3 + 6x^2 - 3x + 7)(x + 2)$:
$= x(4x^3 + 6x^2 - 3x + 7) + 2(4x^3 + 6x^2 - 3x + 7)$
$= 4x^4 + 6x^3 - 3x^2 + 7x + 8x^3 + 12x^2 - 6x + 14$
$= 4x^4 + (6 + 8)x^3 + (-3 + 12)x^2 + (7 - 6)x + 14 = \mathbf{4x^4 + 14x^3 + 9x^2 + x + 14}$.
$= x(4x^3 + 6x^2 - 3x + 7) + 2(4x^3 + 6x^2 - 3x + 7)$
$= 4x^4 + 6x^3 - 3x^2 + 7x + 8x^3 + 12x^2 - 6x + 14$
$= 4x^4 + (6 + 8)x^3 + (-3 + 12)x^2 + (7 - 6)x + 14 = \mathbf{4x^4 + 14x^3 + 9x^2 + x + 14}$.
(viii) $(5x^2 - 6x + 8)(3x + 2)$:
$= 3x(5x^2 - 6x + 8) + 2(5x^2 - 6x + 8)$
$= 15x^3 - 18x^2 + 24x + 10x^2 - 12x + 16$
$= 15x^3 + (-18 + 10)x^2 + (24 - 12)x + 16 = \mathbf{15x^3 - 8x^2 + 12x + 16}$.
$= 3x(5x^2 - 6x + 8) + 2(5x^2 - 6x + 8)$
$= 15x^3 - 18x^2 + 24x + 10x^2 - 12x + 16$
$= 15x^3 + (-18 + 10)x^2 + (24 - 12)x + 16 = \mathbf{15x^3 - 8x^2 + 12x + 16}$.
Question 3
Exercise 6.2
The dimensions of a rectangle are $x + 3$ and $x + 2$. Find the area of the rectangle in term of $x$.
Step-by-Step Working:
1. Formula for area of a rectangle: $\text{Area} = \text{Length} \times \text{Width}$.
2. $\text{Area} = (x + 3)(x + 2)$.
3. Apply distributive law: $x(x + 2) + 3(x + 2) = x^2 + 2x + 3x + 6$.
4. Combine like terms: $x^2 + 5x + 6$.
1. Formula for area of a rectangle: $\text{Area} = \text{Length} \times \text{Width}$.
2. $\text{Area} = (x + 3)(x + 2)$.
3. Apply distributive law: $x(x + 2) + 3(x + 2) = x^2 + 2x + 3x + 6$.
4. Combine like terms: $x^2 + 5x + 6$.
Final Answer: $\mathbf{x^2 + 5x + 6}$ square units.
Question 4
Exercise 6.2
Simplify:
(i) $(x + 2)(x - 5) + (3x + 1)(x - 3)$:
First product: $(x+2)(x-5) = x^2 - 5x + 2x - 10 = x^2 - 3x - 10$.
Second product: $(3x+1)(x-3) = 3x^2 - 9x + x - 3 = 3x^2 - 8x - 3$.
Sum $= (x^2 - 3x - 10) + (3x^2 - 8x - 3) = (1+3)x^2 + (-3-8)x + (-10-3) = \mathbf{4x^2 - 11x - 13}$.
First product: $(x+2)(x-5) = x^2 - 5x + 2x - 10 = x^2 - 3x - 10$.
Second product: $(3x+1)(x-3) = 3x^2 - 9x + x - 3 = 3x^2 - 8x - 3$.
Sum $= (x^2 - 3x - 10) + (3x^2 - 8x - 3) = (1+3)x^2 + (-3-8)x + (-10-3) = \mathbf{4x^2 - 11x - 13}$.
(ii) $(2x - 1)(x + 3) - (4x - 2)(6x + 4)$:
First product: $(2x-1)(x+3) = 2x^2 + 6x - x - 3 = 2x^2 + 5x - 3$.
Second product: $(4x-2)(6x+4) = 24x^2 + 16x - 12x - 8 = 24x^2 + 4x - 8$.
Difference $= (2x^2 + 5x - 3) - (24x^2 + 4x - 8) = 2x^2 + 5x - 3 - 24x^2 - 4x + 8$
$= (2 - 24)x^2 + (5 - 4)x + (-3 + 8) = \mathbf{-22x^2 + x + 5}$.
First product: $(2x-1)(x+3) = 2x^2 + 6x - x - 3 = 2x^2 + 5x - 3$.
Second product: $(4x-2)(6x+4) = 24x^2 + 16x - 12x - 8 = 24x^2 + 4x - 8$.
Difference $= (2x^2 + 5x - 3) - (24x^2 + 4x - 8) = 2x^2 + 5x - 3 - 24x^2 - 4x + 8$
$= (2 - 24)x^2 + (5 - 4)x + (-3 + 8) = \mathbf{-22x^2 + x + 5}$.
Question 5
Exercise 6.2
Find the area and the perimeter of the following figures:
(i) Rectangle with Length $= 2x + 1$, Width $= 3x + 4$:
• Area: $(2x + 1)(3x + 4) = 6x^2 + 8x + 3x + 4 = \mathbf{6x^2 + 11x + 4}$
• Perimeter: $2((2x + 1) + (3x + 4)) = 2(5x + 5) = \mathbf{10x + 10}$.
• Area: $(2x + 1)(3x + 4) = 6x^2 + 8x + 3x + 4 = \mathbf{6x^2 + 11x + 4}$
• Perimeter: $2((2x + 1) + (3x + 4)) = 2(5x + 5) = \mathbf{10x + 10}$.
(ii) Rectangle with Length $= 3x + 5$, Width $= 4x + 7$:
• Area: $(3x + 5)(4x + 7) = 12x^2 + 21x + 20x + 35 = \mathbf{12x^2 + 41x + 35}$
• Perimeter: $2((3x + 5) + (4x + 7)) = 2(7x + 12) = \mathbf{14x + 24}$.
• Area: $(3x + 5)(4x + 7) = 12x^2 + 21x + 20x + 35 = \mathbf{12x^2 + 41x + 35}$
• Perimeter: $2((3x + 5) + (4x + 7)) = 2(7x + 12) = \mathbf{14x + 24}$.
(iii) Rectangle with Length $= 4x - 2$, Width $= 3x + 7$:
• Area: $(4x - 2)(3x + 7) = 12x^2 + 28x - 6x - 14 = \mathbf{12x^2 + 22x - 14}$
• Perimeter: $2((4x - 2) + (3x + 7)) = 2(7x + 5) = \mathbf{14x + 10}$.
• Area: $(4x - 2)(3x + 7) = 12x^2 + 28x - 6x - 14 = \mathbf{12x^2 + 22x - 14}$
• Perimeter: $2((4x - 2) + (3x + 7)) = 2(7x + 5) = \mathbf{14x + 10}$.
3. Standard Algebraic Identities I & II: Squares of Binomials
Theory: Geometric Meaning of $(a+b)^2$ and $(a-b)^2$
An algebraic identity is an equation that remains true for every possible numerical value assigned to its variables. It provides a lightning-fast mathematical shortcut that replaces long multiplication!
Identity I: Square of a Sum
$(a + b)^2 = a^2 + 2ab + b^2$
Geometric Proof: Imagine a large square of side length $(a + b)$. Its total area is $(a + b)^2$. When divided, it splits into:
• A large square of side $a$ (area $a^2$)
• Two identical rectangles of length $a$ and width $b$ (area $2ab$)
• A small square of side $b$ (area $b^2$)
Total area $= a^2 + 2ab + b^2$.
Identity II: Square of a Difference
$(a - b)^2 = a^2 - 2ab + b^2$
Geometric Proof: Start with a large square of side $a$ (area $a^2$). To find the unshaded inner square of side $(a - b)$:
• Subtract two strips of width $b$ (area $-2ab$).
• Because the corner square of side $b$ is subtracted twice, we add back one small square $+b^2$.
Resulting area $= a^2 - 2ab + b^2$.
Mental Arithmetic Shortcuts:
• $102^2 = (100 + 2)^2 = 100^2 + 2(100)(2) + 2^2 = 10000 + 400 + 4 = \mathbf{10404}$.
• $1.98^2 = (2 - 0.02)^2 = 2^2 - 2(2)(0.02) + (0.02)^2 = 4 - 0.08 + 0.0004 = \mathbf{3.9196}$.
• $102^2 = (100 + 2)^2 = 100^2 + 2(100)(2) + 2^2 = 10000 + 400 + 4 = \mathbf{10404}$.
• $1.98^2 = (2 - 0.02)^2 = 2^2 - 2(2)(0.02) + (0.02)^2 = 4 - 0.08 + 0.0004 = \mathbf{3.9196}$.
Exercise 6.3 • 100% Complete Step-by-Step Solved Solutions
Question 1
Exercise 6.3
Fill in the blanks:
(i) $(2a + b)^2 = (2a)^2 + 2(\underline{\quad\mathbf{2a}\quad})(b) + (b)^2$ → Blank is $\mathbf{2a}$.
(ii) $(4x + 6y)^2 = 16x^2 + (\underline{\quad\mathbf{48}\quad})xy + 36y^2$ → $2(4x)(6y) = 48xy$, blank is $\mathbf{48}$.
(iii) $(x^2 + \frac{1}{x^2})^2 = x^4 + \underline{\quad\mathbf{2}\quad} + \frac{1}{x^4}$ → $2(x^2)(\frac{1}{x^2}) = 2$, blank is $\mathbf{2}$.
(iv) $(2a + \frac{1}{2b})^2 = 4a^2 + \underline{\quad\mathbf{\frac{2a}{b}}\quad} + \frac{1}{4b^2}$ → $2(2a)(\frac{1}{2b}) = \frac{2a}{b}$, blank is $\mathbf{\frac{2a}{b}}$.
(v) $(2x - 3y)^2 = (2x)^2 - 2(\underline{\quad\mathbf{2x}\quad})(\underline{\quad\mathbf{3y}\quad}) + (3y)^2$ → Blanks are $\mathbf{(2x)(3y)}$.
(vi) $(4a - \underline{\quad\mathbf{6b}\quad})^2 = (4a)^2 - \underline{\quad\mathbf{48ab}\quad} + (6b)^2$ → Blanks are $\mathbf{6b}$ and $\mathbf{48ab}$.
(vii) $(\underline{\quad\mathbf{2x}\quad} - 4y)^2 = (\underline{\quad\mathbf{2x}\quad})^2 - 16xy + (4y)^2$ → Blanks are $\mathbf{2x}$ and $\mathbf{2x}$.
(viii) $(x - 5)^2 = x^2 - 10x + \underline{\quad\mathbf{25}\quad}$ → $(-5)^2 = 25$, blank is $\mathbf{25}$.
Question 2
Exercise 6.3
Expand the following by using the appropriate formula:
(i) $(2a + 7)^2$:
$= (2a)^2 + 2(2a)(7) + 7^2 = \mathbf{4a^2 + 28a + 49}$.
$= (2a)^2 + 2(2a)(7) + 7^2 = \mathbf{4a^2 + 28a + 49}$.
(ii) $(3x + 1)^2$:
$= (3x)^2 + 2(3x)(1) + 1^2 = \mathbf{9x^2 + 6x + 1}$.
$= (3x)^2 + 2(3x)(1) + 1^2 = \mathbf{9x^2 + 6x + 1}$.
(iii) $(8a + 3b)^2$:
$= (8a)^2 + 2(8a)(3b) + (3b)^2 = \mathbf{64a^2 + 48ab + 9b^2}$.
$= (8a)^2 + 2(8a)(3b) + (3b)^2 = \mathbf{64a^2 + 48ab + 9b^2}$.
(iv) $(x^2 + y^2)^2$:
$= (x^2)^2 + 2(x^2)(y^2) + (y^2)^2 = \mathbf{x^4 + 2x^2y^2 + y^4}$.
$= (x^2)^2 + 2(x^2)(y^2) + (y^2)^2 = \mathbf{x^4 + 2x^2y^2 + y^4}$.
(v) $(3x + 4y)^2$:
$= (3x)^2 + 2(3x)(4y) + (4y)^2 = \mathbf{9x^2 + 24xy + 16y^2}$.
$= (3x)^2 + 2(3x)(4y) + (4y)^2 = \mathbf{9x^2 + 24xy + 16y^2}$.
(vi) $(7x + 8y)^2$:
$= (7x)^2 + 2(7x)(8y) + (8y)^2 = \mathbf{49x^2 + 112xy + 64y^2}$.
$= (7x)^2 + 2(7x)(8y) + (8y)^2 = \mathbf{49x^2 + 112xy + 64y^2}$.
(vii) $(\frac{3}{4}x + \frac{4}{3x})^2$:
$= (\frac{3}{4}x)^2 + 2(\frac{3}{4}x)(\frac{4}{3x}) + (\frac{4}{3x})^2 = \mathbf{\frac{9}{16}x^2 + 2 + \frac{16}{9x^2}}$.
$= (\frac{3}{4}x)^2 + 2(\frac{3}{4}x)(\frac{4}{3x}) + (\frac{4}{3x})^2 = \mathbf{\frac{9}{16}x^2 + 2 + \frac{16}{9x^2}}$.
(viii) $(\frac{2}{3}a + \frac{3}{2}b)^2$:
$= (\frac{2}{3}a)^2 + 2(\frac{2}{3}a)(\frac{3}{2}b) + (\frac{3}{2}b)^2 = \mathbf{\frac{4}{9}a^2 + 2ab + \frac{9}{4}b^2}$.
$= (\frac{2}{3}a)^2 + 2(\frac{2}{3}a)(\frac{3}{2}b) + (\frac{3}{2}b)^2 = \mathbf{\frac{4}{9}a^2 + 2ab + \frac{9}{4}b^2}$.
(ix) $(3a - 7b)^2$:
$= (3a)^2 - 2(3a)(7b) + (7b)^2 = \mathbf{9a^2 - 42ab + 49b^2}$.
$= (3a)^2 - 2(3a)(7b) + (7b)^2 = \mathbf{9a^2 - 42ab + 49b^2}$.
(x) $(3a - \frac{1}{3a})^2$:
$= (3a)^2 - 2(3a)(\frac{1}{3a}) + (\frac{1}{3a})^2 = \mathbf{9a^2 - 2 + \frac{1}{9a^2}}$.
$= (3a)^2 - 2(3a)(\frac{1}{3a}) + (\frac{1}{3a})^2 = \mathbf{9a^2 - 2 + \frac{1}{9a^2}}$.
(xi) $(3x - 11y)^2$:
$= (3x)^2 - 2(3x)(11y) + (11y)^2 = \mathbf{9x^2 - 66xy + 121y^2}$.
$= (3x)^2 - 2(3x)(11y) + (11y)^2 = \mathbf{9x^2 - 66xy + 121y^2}$.
(xii) $(\frac{5}{6}x - \frac{3}{4}y)^2$:
$= (\frac{5}{6}x)^2 - 2(\frac{5}{6}x)(\frac{3}{4}y) + (\frac{3}{4}y)^2 = \mathbf{\frac{25}{36}x^2 - \frac{5}{4}xy + \frac{9}{16}y^2}$.
$= (\frac{5}{6}x)^2 - 2(\frac{5}{6}x)(\frac{3}{4}y) + (\frac{3}{4}y)^2 = \mathbf{\frac{25}{36}x^2 - \frac{5}{4}xy + \frac{9}{16}y^2}$.
(xiii) $(\frac{x}{2} - \frac{3}{4}y)^2$:
$= (\frac{x}{2})^2 - 2(\frac{x}{2})(\frac{3}{4}y) + (\frac{3}{4}y)^2 = \mathbf{\frac{x^2}{4} - \frac{3}{4}xy + \frac{9}{16}y^2}$.
$= (\frac{x}{2})^2 - 2(\frac{x}{2})(\frac{3}{4}y) + (\frac{3}{4}y)^2 = \mathbf{\frac{x^2}{4} - \frac{3}{4}xy + \frac{9}{16}y^2}$.
(xiv) $(\frac{2}{3}x - \frac{3}{2}y)^2$:
$= (\frac{2}{3}x)^2 - 2(\frac{2}{3}x)(\frac{3}{2}y) + (\frac{3}{2}y)^2 = \mathbf{\frac{4}{9}x^2 - 2xy + \frac{9}{4}y^2}$.
$= (\frac{2}{3}x)^2 - 2(\frac{2}{3}x)(\frac{3}{2}y) + (\frac{3}{2}y)^2 = \mathbf{\frac{4}{9}x^2 - 2xy + \frac{9}{4}y^2}$.
Question 3
Exercise 6.3
Simplify the following:
(i) $(a + b)^2 + (2a + 2b)^2$:
$= (a^2 + 2ab + b^2) + (4a^2 + 8ab + 4b^2) = (1+4)a^2 + (2+8)ab + (1+4)b^2 = \mathbf{5a^2 + 10ab + 5b^2}$.
$= (a^2 + 2ab + b^2) + (4a^2 + 8ab + 4b^2) = (1+4)a^2 + (2+8)ab + (1+4)b^2 = \mathbf{5a^2 + 10ab + 5b^2}$.
(ii) $(2a + 4b)^2 - (a + 3b)^2$:
$= (4a^2 + 16ab + 16b^2) - (a^2 + 6ab + 9b^2)$
$= 4a^2 + 16ab + 16b^2 - a^2 - 6ab - 9b^2 = \mathbf{3a^2 + 10ab + 7b^2}$.
$= (4a^2 + 16ab + 16b^2) - (a^2 + 6ab + 9b^2)$
$= 4a^2 + 16ab + 16b^2 - a^2 - 6ab - 9b^2 = \mathbf{3a^2 + 10ab + 7b^2}$.
(iii) $(3x + 4y)^2 - (2x + 3y)^2$:
$= (9x^2 + 24xy + 16y^2) - (4x^2 + 12xy + 9y^2)$
$= 9x^2 + 24xy + 16y^2 - 4x^2 - 12xy - 9y^2 = \mathbf{5x^2 + 12xy + 7y^2}$.
$= (9x^2 + 24xy + 16y^2) - (4x^2 + 12xy + 9y^2)$
$= 9x^2 + 24xy + 16y^2 - 4x^2 - 12xy - 9y^2 = \mathbf{5x^2 + 12xy + 7y^2}$.
(iv) $(3x - 4y)^2 + (2x - 3y)^2$:
$= (9x^2 - 24xy + 16y^2) + (4x^2 - 12xy + 9y^2) = (9+4)x^2 + (-24-12)xy + (16+9)y^2 = \mathbf{13x^2 - 36xy + 25y^2}$.
$= (9x^2 - 24xy + 16y^2) + (4x^2 - 12xy + 9y^2) = (9+4)x^2 + (-24-12)xy + (16+9)y^2 = \mathbf{13x^2 - 36xy + 25y^2}$.
(v) $(5x - 4y)^2 + (4x - 2y)^2$:
$= (25x^2 - 40xy + 16y^2) + (16x^2 - 16xy + 4y^2) = (25+16)x^2 + (-40-16)xy + (16+4)y^2 = \mathbf{41x^2 - 56xy + 20y^2}$.
$= (25x^2 - 40xy + 16y^2) + (16x^2 - 16xy + 4y^2) = (25+16)x^2 + (-40-16)xy + (16+4)y^2 = \mathbf{41x^2 - 56xy + 20y^2}$.
(vi) $(8x - 9y)^2 + (6x - 4y)^2$:
$= (64x^2 - 144xy + 81y^2) + (36x^2 - 48xy + 16y^2) = (64+36)x^2 + (-144-48)xy + (81+16)y^2 = \mathbf{100x^2 - 192xy + 97y^2}$.
$= (64x^2 - 144xy + 81y^2) + (36x^2 - 48xy + 16y^2) = (64+36)x^2 + (-144-48)xy + (81+16)y^2 = \mathbf{100x^2 - 192xy + 97y^2}$.
Question 4
Exercise 6.3
Evaluate by using formula:
(i) $(48)^2$:
$= (50 - 2)^2 = 50^2 - 2(50)(2) + 2^2$
$= 2500 - 200 + 4 = \mathbf{2304}$.
$= (50 - 2)^2 = 50^2 - 2(50)(2) + 2^2$
$= 2500 - 200 + 4 = \mathbf{2304}$.
(ii) $(103)^2$:
$= (100 + 3)^2 = 100^2 + 2(100)(3) + 3^2$
$= 10000 + 600 + 9 = \mathbf{10609}$.
$= (100 + 3)^2 = 100^2 + 2(100)(3) + 3^2$
$= 10000 + 600 + 9 = \mathbf{10609}$.
(iii) $(1.96)^2$:
$= (2 - 0.04)^2 = 2^2 - 2(2)(0.04) + (0.04)^2$
$= 4 - 0.16 + 0.0016 = \mathbf{3.8416}$.
$= (2 - 0.04)^2 = 2^2 - 2(2)(0.04) + (0.04)^2$
$= 4 - 0.16 + 0.0016 = \mathbf{3.8416}$.
(iv) $(504)^2$:
$= (500 + 4)^2 = 500^2 + 2(500)(4) + 4^2$
$= 250000 + 4000 + 16 = \mathbf{254016}$.
$= (500 + 4)^2 = 500^2 + 2(500)(4) + 4^2$
$= 250000 + 4000 + 16 = \mathbf{254016}$.
(v) $(999)^2$:
$= (1000 - 1)^2 = 1000^2 - 2(1000)(1) + 1^2$
$= 1000000 - 2000 + 1 = \mathbf{998001}$.
$= (1000 - 1)^2 = 1000^2 - 2(1000)(1) + 1^2$
$= 1000000 - 2000 + 1 = \mathbf{998001}$.
(vi) $(7.03)^2$:
$= (7 + 0.03)^2 = 7^2 + 2(7)(0.03) + (0.03)^2$
$= 49 + 0.42 + 0.0009 = \mathbf{49.4209}$.
$= (7 + 0.03)^2 = 7^2 + 2(7)(0.03) + (0.03)^2$
$= 49 + 0.42 + 0.0009 = \mathbf{49.4209}$.
(vii) $(5.2)^2$:
$= (5 + 0.2)^2 = 5^2 + 2(5)(0.2) + (0.2)^2$
$= 25 + 2.0 + 0.04 = \mathbf{27.04}$.
$= (5 + 0.2)^2 = 5^2 + 2(5)(0.2) + (0.2)^2$
$= 25 + 2.0 + 0.04 = \mathbf{27.04}$.
(viii) $(2.01)^2$:
$= (2 + 0.01)^2 = 2^2 + 2(2)(0.01) + (0.01)^2$
$= 4 + 0.04 + 0.0001 = \mathbf{4.0401}$.
$= (2 + 0.01)^2 = 2^2 + 2(2)(0.01) + (0.01)^2$
$= 4 + 0.04 + 0.0001 = \mathbf{4.0401}$.
(ix) $(9.20)^2$:
$= (9 + 0.2)^2 = 9^2 + 2(9)(0.2) + (0.2)^2$
$= 81 + 3.6 + 0.04 = \mathbf{84.64}$.
$= (9 + 0.2)^2 = 9^2 + 2(9)(0.2) + (0.2)^2$
$= 81 + 3.6 + 0.04 = \mathbf{84.64}$.
4. Standard Algebraic Identities III & IV: Cross-Products & Difference of Two Squares
Theory: Cross Products & The Difference of Two Squares
Identity III: Product of $(x + a)$ and $(x + b)$
$(x + a)(x + b) = x^2 + (a + b)x + ab$
Proof: $x(x + b) + a(x + b) = x^2 + bx + ax + ab = x^2 + (a + b)x + ab$.
Memory Hook: The middle coefficient is the sum of constants $(a + b)$, and the last term is the product $(ab)$.
Identity IV: Difference of Two Squares
$(a + b)(a - b) = a^2 - b^2$
Proof: $a(a - b) + b(a - b) = a^2 - ab + ba - b^2 = a^2 - b^2$.
Geometric Visual: Cutting out a square of side $b$ from a larger square of side $a$ leaves area $a^2 - b^2$. Rearranging the remaining pieces forms a single rectangle of length $(a + b)$ and width $(a - b)$!
Mental Arithmetic Shortcuts:
• $103 \times 96 = (100 + 3)(100 - 4) = 100^2 + (3 - 4)(100) + (3)(-4) = 10000 - 100 - 12 = \mathbf{9888}$.
• $52 \times 48 = (50 + 2)(50 - 2) = 50^2 - 2^2 = 2500 - 4 = \mathbf{2496}$.
• $102 \times 98 = (100 + 2)(100 - 2) = 100^2 - 2^2 = 10000 - 4 = \mathbf{9996}$.
• $103 \times 96 = (100 + 3)(100 - 4) = 100^2 + (3 - 4)(100) + (3)(-4) = 10000 - 100 - 12 = \mathbf{9888}$.
• $52 \times 48 = (50 + 2)(50 - 2) = 50^2 - 2^2 = 2500 - 4 = \mathbf{2496}$.
• $102 \times 98 = (100 + 2)(100 - 2) = 100^2 - 2^2 = 10000 - 4 = \mathbf{9996}$.
Exercise 6.4 • 100% Complete Step-by-Step Solved Solutions
Question 1
Exercise 6.4
Find the following products by using the appropriate formula $(x + a)(x + b) = x^2 + (a + b)x + ab$:
(i) $(x + 3)(x + 4)$:
$= x^2 + (3 + 4)x + (3 \times 4) = \mathbf{x^2 + 7x + 12}$.
$= x^2 + (3 + 4)x + (3 \times 4) = \mathbf{x^2 + 7x + 12}$.
(ii) $(x + 5)(x + 7)$:
$= x^2 + (5 + 7)x + (5 \times 7) = \mathbf{x^2 + 12x + 35}$.
$= x^2 + (5 + 7)x + (5 \times 7) = \mathbf{x^2 + 12x + 35}$.
(iii) $(p - 4)(p + 6)$:
$= p^2 + (-4 + 6)p + (-4 \times 6) = \mathbf{p^2 + 2p - 24}$.
$= p^2 + (-4 + 6)p + (-4 \times 6) = \mathbf{p^2 + 2p - 24}$.
(iv) $(z + 7)(z + 9)$:
$= z^2 + (7 + 9)z + (7 \times 9) = \mathbf{z^2 + 16z + 63}$.
$= z^2 + (7 + 9)z + (7 \times 9) = \mathbf{z^2 + 16z + 63}$.
(v) $(2x - 4)(2x + 5)$:
$= (2x)^2 + (-4 + 5)(2x) + (-4 \times 5) = 4x^2 + 1(2x) - 20 = \mathbf{4x^2 + 2x - 20}$.
$= (2x)^2 + (-4 + 5)(2x) + (-4 \times 5) = 4x^2 + 1(2x) - 20 = \mathbf{4x^2 + 2x - 20}$.
(vi) $(3x + 7)(3x - 2)$:
$= (3x)^2 + (7 - 2)(3x) + (7 \times -2) = 9x^2 + 5(3x) - 14 = \mathbf{9x^2 + 15x - 14}$.
$= (3x)^2 + (7 - 2)(3x) + (7 \times -2) = 9x^2 + 5(3x) - 14 = \mathbf{9x^2 + 15x - 14}$.
Question 2
Exercise 6.4
Evaluate the following by using formula:
(i) $103 \times 96$:
$= (100 + 3)(100 - 4)$
$= 100^2 + (3 - 4)(100) + (3)(-4)$
$= 10000 - 100 - 12 = \mathbf{9888}$.
$= (100 + 3)(100 - 4)$
$= 100^2 + (3 - 4)(100) + (3)(-4)$
$= 10000 - 100 - 12 = \mathbf{9888}$.
(ii) $104 \times 105$:
$= (100 + 4)(100 + 5)$
$= 100^2 + (4 + 5)(100) + (4 \times 5)$
$= 10000 + 900 + 20 = \mathbf{10920}$.
$= (100 + 4)(100 + 5)$
$= 100^2 + (4 + 5)(100) + (4 \times 5)$
$= 10000 + 900 + 20 = \mathbf{10920}$.
(iii) $998 \times 1002$:
$= (1000 - 2)(1000 + 2) = 1000^2 - 2^2$
$= 1000000 - 4 = \mathbf{999996}$.
$= (1000 - 2)(1000 + 2) = 1000^2 - 2^2$
$= 1000000 - 4 = \mathbf{999996}$.
Question 3
Exercise 6.4
Find the product without actual multiplication using $(a + b)(a - b) = a^2 - b^2$:
(i) $(x + 4)(x - 4)$:
$= (x)^2 - (4)^2 = \mathbf{x^2 - 16}$.
$= (x)^2 - (4)^2 = \mathbf{x^2 - 16}$.
(ii) $(7x + 8)(7x - 8)$:
$= (7x)^2 - (8)^2 = \mathbf{49x^2 - 64}$.
$= (7x)^2 - (8)^2 = \mathbf{49x^2 - 64}$.
(iii) $(6x + \frac{3}{8})(6x - \frac{3}{8})$:
$= (6x)^2 - (\frac{3}{8})^2 = \mathbf{36x^2 - \frac{9}{64}}$.
$= (6x)^2 - (\frac{3}{8})^2 = \mathbf{36x^2 - \frac{9}{64}}$.
(iv) $(x + 2y)(x - 2y)$:
$= (x)^2 - (2y)^2 = \mathbf{x^2 - 4y^2}$.
$= (x)^2 - (2y)^2 = \mathbf{x^2 - 4y^2}$.
(v) $(4x + 7y)(4x - 7y)$:
$= (4x)^2 - (7y)^2 = \mathbf{16x^2 - 49y^2}$.
$= (4x)^2 - (7y)^2 = \mathbf{16x^2 - 49y^2}$.
(vi) $(2a + 9b)(2a - 9b)$:
$= (2a)^2 - (9b)^2 = \mathbf{4a^2 - 81b^2}$.
$= (2a)^2 - (9b)^2 = \mathbf{4a^2 - 81b^2}$.
Question 4
Exercise 6.4
Find the continuous product of the following:
(i) $(a + b)(a - b)(a^2 + b^2)$:
Step 1: $(a + b)(a - b) = a^2 - b^2$.
Step 2: $(a^2 - b^2)(a^2 + b^2) = (a^2)^2 - (b^2)^2 = \mathbf{a^4 - b^4}$.
Step 1: $(a + b)(a - b) = a^2 - b^2$.
Step 2: $(a^2 - b^2)(a^2 + b^2) = (a^2)^2 - (b^2)^2 = \mathbf{a^4 - b^4}$.
(ii) $(x + 2y)(x - 2y)(x^2 + 4y^2)$:
Step 1: $(x + 2y)(x - 2y) = x^2 - 4y^2$.
Step 2: $(x^2 - 4y^2)(x^2 + 4y^2) = (x^2)^2 - (4y^2)^2 = \mathbf{x^4 - 16y^4}$.
Step 1: $(x + 2y)(x - 2y) = x^2 - 4y^2$.
Step 2: $(x^2 - 4y^2)(x^2 + 4y^2) = (x^2)^2 - (4y^2)^2 = \mathbf{x^4 - 16y^4}$.
(iii) $(4a + b)(4a - b)(16a^2 + b^2)$:
Step 1: $(4a + b)(4a - b) = 16a^2 - b^2$.
Step 2: $(16a^2 - b^2)(16a^2 + b^2) = (16a^2)^2 - (b^2)^2 = \mathbf{256a^4 - b^4}$.
Step 1: $(4a + b)(4a - b) = 16a^2 - b^2$.
Step 2: $(16a^2 - b^2)(16a^2 + b^2) = (16a^2)^2 - (b^2)^2 = \mathbf{256a^4 - b^4}$.
(iv) $(a + 3)(a - 3)(a^2 + 9)$:
Step 1: $(a + 3)(a - 3) = a^2 - 9$.
Step 2: $(a^2 - 9)(a^2 + 9) = (a^2)^2 - (9)^2 = \mathbf{a^4 - 81}$.
Step 1: $(a + 3)(a - 3) = a^2 - 9$.
Step 2: $(a^2 - 9)(a^2 + 9) = (a^2)^2 - (9)^2 = \mathbf{a^4 - 81}$.
Question 5
Exercise 6.4
Evaluate with the help of formula:
(i) $102 \times 98$:
$= (100 + 2)(100 - 2) = 100^2 - 2^2$
$= 10000 - 4 = \mathbf{9996}$.
$= (100 + 2)(100 - 2) = 100^2 - 2^2$
$= 10000 - 4 = \mathbf{9996}$.
(ii) $65 \times 55$:
$= (60 + 5)(60 - 5) = 60^2 - 5^2$
$= 3600 - 25 = \mathbf{3575}$.
$= (60 + 5)(60 - 5) = 60^2 - 5^2$
$= 3600 - 25 = \mathbf{3575}$.
(iii) $(1.01) \times (0.99)$:
$= (1 + 0.01)(1 - 0.01) = 1^2 - (0.01)^2$
$= 1 - 0.0001 = \mathbf{0.9999}$.
$= (1 + 0.01)(1 - 0.01) = 1^2 - (0.01)^2$
$= 1 - 0.0001 = \mathbf{0.9999}$.
(iv) $202 \times 198$:
$= (200 + 2)(200 - 2) = 200^2 - 2^2$
$= 40000 - 4 = \mathbf{39996}$.
$= (200 + 2)(200 - 2) = 200^2 - 2^2$
$= 40000 - 4 = \mathbf{39996}$.
Question 6
Exercise 6.4
Simplify the following:
(i) $(x + 2)(x - 2) + (x + 2)^2$:
First term: $(x + 2)(x - 2) = x^2 - 4$.
Second term: $(x + 2)^2 = x^2 + 4x + 4$.
Sum $= (x^2 - 4) + (x^2 + 4x + 4) = (1 + 1)x^2 + 4x + (-4 + 4) = \mathbf{2x^2 + 4x}$.
First term: $(x + 2)(x - 2) = x^2 - 4$.
Second term: $(x + 2)^2 = x^2 + 4x + 4$.
Sum $= (x^2 - 4) + (x^2 + 4x + 4) = (1 + 1)x^2 + 4x + (-4 + 4) = \mathbf{2x^2 + 4x}$.
(ii) $(3a - 2)(3a + 2) - (a - 4)^2$:
First term: $(3a - 2)(3a + 2) = 9a^2 - 4$.
Second term: $(a - 4)^2 = a^2 - 8a + 16$.
Difference $= (9a^2 - 4) - (a^2 - 8a + 16) = 9a^2 - 4 - a^2 + 8a - 16$
$= (9 - 1)a^2 + 8a + (-4 - 16) = \mathbf{8a^2 + 8a - 20}$.
First term: $(3a - 2)(3a + 2) = 9a^2 - 4$.
Second term: $(a - 4)^2 = a^2 - 8a + 16$.
Difference $= (9a^2 - 4) - (a^2 - 8a + 16) = 9a^2 - 4 - a^2 + 8a - 16$
$= (9 - 1)a^2 + 8a + (-4 - 16) = \mathbf{8a^2 + 8a - 20}$.
(iii) $(2x - y)(2x + y) - (x + 2y)(x - 2y)$:
First term: $(2x - y)(2x + y) = 4x^2 - y^2$.
Second term: $(x + 2y)(x - 2y) = x^2 - 4y^2$.
Difference $= (4x^2 - y^2) - (x^2 - 4y^2) = 4x^2 - y^2 - x^2 + 4y^2$
$= (4 - 1)x^2 + (-1 + 4)y^2 = \mathbf{3x^2 + 3y^2}$.
First term: $(2x - y)(2x + y) = 4x^2 - y^2$.
Second term: $(x + 2y)(x - 2y) = x^2 - 4y^2$.
Difference $= (4x^2 - y^2) - (x^2 - 4y^2) = 4x^2 - y^2 - x^2 + 4y^2$
$= (4 - 1)x^2 + (-1 + 4)y^2 = \mathbf{3x^2 + 3y^2}$.
5. Factorization of Algebraic Expressions (Exercises 6.5 & 6.6)
Theory: The Three Pillars of Factorization
In arithmetic, factoring $12$ means writing it as $3 \times 4$. In algebra, factorization is the process of writing an algebraic expression as the product of its irreducible factors. It is the exact inverse (undoing) of multiplication!
Method 1: Common Monomial Factor
$ka + kb + kc = k(a + b + c)$
Inspect all terms to find the Greatest Common Factor (GCF) of numerical coefficients and the lowest power of every shared variable. Factor the GCF outside parentheses. E.g.: $8a^4 + 12a^2 = 4a^2(2a^2 + 3)$.
Method 2: Grouping 4 Terms
$ac + ad + bc + bd = (a + b)(c + d)$
Pair terms with common factors into two groups of two: $a(c + d) + b(c + d)$. Then pull out the identical binomial $(c + d)$ as a common factor: $(c + d)(a + b)$.
Method 3: Middle Term Splitting ($ax^2 + bx + c$)
Product $= ac$, Sum $= b$
Find two numbers $p$ and $q$ whose product is $a \times c$ and whose algebraic sum is $b$. Split $bx$ into $px + qx$ and apply the grouping method!
Exercise 6.5 • 100% Complete Step-by-Step Solved Solutions
Questions 1 to 16: Factorization
Exercise 6.5
1. $x^2 + x$:
Common factor is $x$: $x(x) + x(1) = \mathbf{x(x + 1)}$.
Common factor is $x$: $x(x) + x(1) = \mathbf{x(x + 1)}$.
2. $x^3 + x^2$:
Common factor is $x^2$: $x^2(x) + x^2(1) = \mathbf{x^2(x + 1)}$.
Common factor is $x^2$: $x^2(x) + x^2(1) = \mathbf{x^2(x + 1)}$.
3. $2y^3 + 4y$:
Common factor is $2y$: $2y(y^2) + 2y(2) = \mathbf{2y(y^2 + 2)}$.
Common factor is $2y$: $2y(y^2) + 2y(2) = \mathbf{2y(y^2 + 2)}$.
4. $4a^4 - 20a^2$:
Common factor is $4a^2$: $4a^2(a^2) - 4a^2(5) = \mathbf{4a^2(a^2 - 5)}$.
Common factor is $4a^2$: $4a^2(a^2) - 4a^2(5) = \mathbf{4a^2(a^2 - 5)}$.
5. $2y^3 + 7y^2 + y$:
Common factor is $y$: $y(2y^2 + 7y + 1) = \mathbf{y(2y^2 + 7y + 1)}$.
Common factor is $y$: $y(2y^2 + 7y + 1) = \mathbf{y(2y^2 + 7y + 1)}$.
6. $5a^4 - 4a^2 + 3a$:
Common factor is $a$: $a(5a^3 - 4a + 3) = \mathbf{a(5a^3 - 4a + 3)}$.
Common factor is $a$: $a(5a^3 - 4a + 3) = \mathbf{a(5a^3 - 4a + 3)}$.
7. $xy - yz^2 + x^2y^2z^2$:
Common factor is $y$: $y(x - z^2 + x^2yz^2) = \mathbf{y(x - z^2 + x^2yz^2)}$.
Common factor is $y$: $y(x - z^2 + x^2yz^2) = \mathbf{y(x - z^2 + x^2yz^2)}$.
8. $7a - 7a^3 + 14a^4$:
Common factor is $7a$: $7a(1 - a^2 + 2a^3) = \mathbf{7a(1 - a^2 + 2a^3)}$.
Common factor is $7a$: $7a(1 - a^2 + 2a^3) = \mathbf{7a(1 - a^2 + 2a^3)}$.
9. $x^2y - 2x^2y + x^3y^2$:
Combine like terms first: $-x^2y + x^3y^2 = \mathbf{x^2y(xy - 1)}$ (or $x^2y(1 - 2 + xy)$).
Combine like terms first: $-x^2y + x^3y^2 = \mathbf{x^2y(xy - 1)}$ (or $x^2y(1 - 2 + xy)$).
10. $2x^2y^3 - 6x^2y^2 + 2xy^3$:
Common factor is $2xy^2$: $2xy^2(xy - 3x + y) = \mathbf{2xy^2(xy - 3x + y)}$.
Common factor is $2xy^2$: $2xy^2(xy - 3x + y) = \mathbf{2xy^2(xy - 3x + y)}$.
11. $2x^3y^3z^4 - 4x^2y^2z^3 + 3x^3y^2z^4$:
Common factor is $x^2y^2z^3$: $\mathbf{x^2y^2z^3(2xyz - 4 + 3xz)}$.
Common factor is $x^2y^2z^3$: $\mathbf{x^2y^2z^3(2xyz - 4 + 3xz)}$.
12. $a^2 + ab + ac + bc$:
Group pairs: $a(a + b) + c(a + b) = \mathbf{(a + b)(a + c)}$.
Group pairs: $a(a + b) + c(a + b) = \mathbf{(a + b)(a + c)}$.
13. $x^3 - x^2y^2 - x^3y$:
Common factor is $x^2$: $\mathbf{x^2(x - y^2 - xy)}$.
Common factor is $x^2$: $\mathbf{x^2(x - y^2 - xy)}$.
14. $t^2 + 4t - st - 4s$:
Group pairs: $t(t + 4) - s(t + 4) = \mathbf{(t + 4)(t - s)}$.
Group pairs: $t(t + 4) - s(t + 4) = \mathbf{(t + 4)(t - s)}$.
15. $x^2(x - y) - y^2(x - y) + z^2(x - y)$:
Take $(x - y)$ common: $\mathbf{(x - y)(x^2 - y^2 + z^2)}$.
Take $(x - y)$ common: $\mathbf{(x - y)(x^2 - y^2 + z^2)}$.
16. $5x^2(x + y) + 10xy(x + y) + 25xy^2(x + y)$:
Take $5x(x + y)$ common: $\mathbf{5x(x + y)(x + 2y + 5y^2)}$.
Take $5x(x + y)$ common: $\mathbf{5x(x + y)(x + 2y + 5y^2)}$.
Questions 17 & 18: Real-World Geometric Factorization
Exercise 6.5
Question 17: A designer preparing to paint a wall found that its area is $3m(m + 7) + k(m + 7)$. Find the length and width of the rectangle.
Solution:
1. $\text{Area} = \text{Length} \times \text{Width}$.
2. Take the common binomial factor $(m + 7)$ out:
$\text{Area} = (3m + k)(m + 7)$.
Final Answer: $\mathbf{\text{Length} = 3m + k}$, $\mathbf{\text{Width} = m + 7}$ (or vice versa).
Solution:
1. $\text{Area} = \text{Length} \times \text{Width}$.
2. Take the common binomial factor $(m + 7)$ out:
$\text{Area} = (3m + k)(m + 7)$.
Final Answer: $\mathbf{\text{Length} = 3m + k}$, $\mathbf{\text{Width} = m + 7}$ (or vice versa).
Question 18: Area of the rectangle is $24a^2b - 18ab^2$. Find its length and width.
Solution:
1. $\text{Area} = 24a^2b - 18ab^2$.
2. Find the GCF of coefficients: $\text{GCF}(24, 18) = 6$.
3. Shared variables with lowest powers: $a$ and $b$. Thus GCF $= 6ab$.
4. Factor out $6ab$: $6ab(4a - 3b)$.
Final Answer: $\mathbf{\text{Length} = 6ab}$, $\mathbf{\text{Width} = 4a - 3b}$ (or vice versa).
Solution:
1. $\text{Area} = 24a^2b - 18ab^2$.
2. Find the GCF of coefficients: $\text{GCF}(24, 18) = 6$.
3. Shared variables with lowest powers: $a$ and $b$. Thus GCF $= 6ab$.
4. Factor out $6ab$: $6ab(4a - 3b)$.
Final Answer: $\mathbf{\text{Length} = 6ab}$, $\mathbf{\text{Width} = 4a - 3b}$ (or vice versa).
Exercise 6.6 • 100% Complete Step-by-Step Solved Solutions
Questions 1 to 10: Splitting the Middle Term
Exercise 6.6
1. $x^2 + 9x + 18$:
Product $= 18$, Sum $= 9$ → Factors: $6$ and $3$.
$= x^2 + 6x + 3x + 18 = x(x + 6) + 3(x + 6) = \mathbf{(x + 6)(x + 3)}$.
Product $= 18$, Sum $= 9$ → Factors: $6$ and $3$.
$= x^2 + 6x + 3x + 18 = x(x + 6) + 3(x + 6) = \mathbf{(x + 6)(x + 3)}$.
2. $x^2 - 9x + 8$:
Product $= 8$, Sum $= -9$ → Factors: $-1$ and $-8$.
$= x^2 - x - 8x + 8 = x(x - 1) - 8(x - 1) = \mathbf{(x - 1)(x - 8)}$.
Product $= 8$, Sum $= -9$ → Factors: $-1$ and $-8$.
$= x^2 - x - 8x + 8 = x(x - 1) - 8(x - 1) = \mathbf{(x - 1)(x - 8)}$.
3. $x^2 - 5x - 6$:
Product $= -6$, Sum $= -5$ → Factors: $-6$ and $+1$.
$= x^2 - 6x + x - 6 = x(x - 6) + 1(x - 6) = \mathbf{(x - 6)(x + 1)}$.
Product $= -6$, Sum $= -5$ → Factors: $-6$ and $+1$.
$= x^2 - 6x + x - 6 = x(x - 6) + 1(x - 6) = \mathbf{(x - 6)(x + 1)}$.
4. $x^2 - 12x + 35$:
Product $= 35$, Sum $= -12$ → Factors: $-7$ and $-5$.
$= x^2 - 7x - 5x + 35 = x(x - 7) - 5(x - 7) = \mathbf{(x - 7)(x - 5)}$.
Product $= 35$, Sum $= -12$ → Factors: $-7$ and $-5$.
$= x^2 - 7x - 5x + 35 = x(x - 7) - 5(x - 7) = \mathbf{(x - 7)(x - 5)}$.
5. $x^2 + 10x + 21$:
Product $= 21$, Sum $= 10$ → Factors: $7$ and $3$.
$= x^2 + 7x + 3x + 21 = x(x + 7) + 3(x + 7) = \mathbf{(x + 7)(x + 3)}$.
Product $= 21$, Sum $= 10$ → Factors: $7$ and $3$.
$= x^2 + 7x + 3x + 21 = x(x + 7) + 3(x + 7) = \mathbf{(x + 7)(x + 3)}$.
6. $2a^2 - 4a - 6$:
Factor out 2: $2(a^2 - 2a - 3)$. Product $= -3$, Sum $= -2$ → $-3, +1$.
$= 2[a(a - 3) + 1(a - 3)] = \mathbf{2(a + 1)(a - 3)}$.
Factor out 2: $2(a^2 - 2a - 3)$. Product $= -3$, Sum $= -2$ → $-3, +1$.
$= 2[a(a - 3) + 1(a - 3)] = \mathbf{2(a + 1)(a - 3)}$.
7. $x^2 - 21x + 90$:
Product $= 90$, Sum $= -21$ → Factors: $-6$ and $-15$.
$= x^2 - 6x - 15x + 90 = x(x - 6) - 15(x - 6) = \mathbf{(x - 6)(x - 15)}$.
Product $= 90$, Sum $= -21$ → Factors: $-6$ and $-15$.
$= x^2 - 6x - 15x + 90 = x(x - 6) - 15(x - 6) = \mathbf{(x - 6)(x - 15)}$.
8. $x^2 + x - 2$:
Product $= -2$, Sum $= +1$ → Factors: $+2$ and $-1$.
$= x^2 + 2x - x - 2 = x(x + 2) - 1(x + 2) = \mathbf{(x - 1)(x + 2)}$.
Product $= -2$, Sum $= +1$ → Factors: $+2$ and $-1$.
$= x^2 + 2x - x - 2 = x(x + 2) - 1(x + 2) = \mathbf{(x - 1)(x + 2)}$.
9. $t^2 + 12t + 35$:
Product $= 35$, Sum $= 12$ → Factors: $5$ and $7$.
$= t^2 + 5t + 7t + 35 = t(t + 5) + 7(t + 5) = \mathbf{(t + 5)(t + 7)}$.
Product $= 35$, Sum $= 12$ → Factors: $5$ and $7$.
$= t^2 + 5t + 7t + 35 = t(t + 5) + 7(t + 5) = \mathbf{(t + 5)(t + 7)}$.
10. $3y^2 + 5y + 2$:
Product $= 3 \times 2 = 6$, Sum $= 5$ → Factors: $3$ and $2$.
$= 3y^2 + 3y + 2y + 2 = 3y(y + 1) + 2(y + 1) = \mathbf{(3y + 2)(y + 1)}$.
Product $= 3 \times 2 = 6$, Sum $= 5$ → Factors: $3$ and $2$.
$= 3y^2 + 3y + 2y + 2 = 3y(y + 1) + 2(y + 1) = \mathbf{(3y + 2)(y + 1)}$.
6. Number Patterns, Sequences & The Fibonacci Miracle
Theory: The Language of Patterns
A sequence is an ordered list of numbers formed by applying a definite mathematical rule. The individual numbers in the sequence are called its terms.
Arithmetic Sequence
Add/Subtract Constant ($d$)
Each next term is obtained by adding a fixed number $d$ (common difference) to the previous term.
General Term Formula: $T_n = a + (n - 1)d$, where $a$ is the first term and $n$ is the term position. E.g., $12, 17, 22, 27, \dots$ ($a=12, d=5$).
Geometric Sequence
Multiply/Divide Constant ($r$)
Each next term is obtained by multiplying the previous term by a fixed ratio $r$.
E.g., $1, 2, 4, 8, 16, 32, \dots$ (multiply by $2$).
$88, 44, 22, 11, \dots$ (divide by $2$ or multiply by $\\frac{1}{2}$).
The Fibonacci Sequence
$1, 1, 2, 3, 5, 8, 13, 21, 34, 55, \dots$
Named after Italian mathematician Leonardo Fibonacci. Starting with $1, 1$, every subsequent term is the sum of the two preceding terms: $1+1=2, 1+2=3, 2+3=5, 3+5=8, \dots$
In Nature: The seeds of a sunflower spiral in alternating Fibonacci pairs (e.g. 21 spirals clockwise and 34 counter-clockwise). Pine cones display 5 and 8 spirals!
Exercise 6.7 • 100% Complete Step-by-Step Solved Solutions
Question 1
Exercise 6.7
Describe the pattern in each sequence then find the next three terms:
(i) $7, 14, 21, 28, 35, 42, \dots$:
Rule: Add $7$ to previous term (multiples of $7$).
Next three terms: $\mathbf{49, 56, 63}$.
Rule: Add $7$ to previous term (multiples of $7$).
Next three terms: $\mathbf{49, 56, 63}$.
(ii) $2, 6, 18, 54, \dots$:
Rule: Multiply previous term by $3$.
Next three terms: $54 \times 3 = 162$, $162 \times 3 = 486$, $486 \times 3 = 1458$ → $\mathbf{162, 486, 1458}$.
Rule: Multiply previous term by $3$.
Next three terms: $54 \times 3 = 162$, $162 \times 3 = 486$, $486 \times 3 = 1458$ → $\mathbf{162, 486, 1458}$.
(iii) $0, 1, 3, 6, 10, 15, \dots$:
Rule: Add consecutive integers $+1, +2, +3, +4, +5, \dots$
Next three terms: $15+6 = 21$, $21+7 = 28$, $28+8 = 36$ → $\mathbf{21, 28, 36}$.
Rule: Add consecutive integers $+1, +2, +3, +4, +5, \dots$
Next three terms: $15+6 = 21$, $21+7 = 28$, $28+8 = 36$ → $\mathbf{21, 28, 36}$.
(iv) $15, 30, 45, 60, \dots$:
Rule: Add $15$ to previous term.
Next three terms: $\mathbf{75, 90, 105}$.
Rule: Add $15$ to previous term.
Next three terms: $\mathbf{75, 90, 105}$.
(v) $9, 3, 1, \frac{1}{3}, \frac{1}{9}, \dots$:
Rule: Divide by $3$ (multiply by $\frac{1}{3}$).
Next three terms: $\mathbf{\frac{1}{27}, \frac{1}{81}, \frac{1}{243}}$.
Rule: Divide by $3$ (multiply by $\frac{1}{3}$).
Next three terms: $\mathbf{\frac{1}{27}, \frac{1}{81}, \frac{1}{243}}$.
(vi) $15, 17, 19, \dots$:
Rule: Add $2$ to previous term.
Next three terms: $\mathbf{21, 23, 25}$.
Rule: Add $2$ to previous term.
Next three terms: $\mathbf{21, 23, 25}$.
(vii) $87, 82, 77, \dots$:
Rule: Subtract $5$ from previous term.
Next three terms: $\mathbf{72, 67, 62}$.
Rule: Subtract $5$ from previous term.
Next three terms: $\mathbf{72, 67, 62}$.
(viii) $88, 44, 22, \dots$:
Rule: Divide by $2$ (multiply by $\frac{1}{2}$).
Next three terms: $22 \div 2 = 11$, $11 \div 2 = \frac{11}{2}$, $\frac{11}{2} \div 2 = \frac{11}{4}$ → $\mathbf{11, \frac{11}{2}, \frac{11}{4}}$.
Rule: Divide by $2$ (multiply by $\frac{1}{2}$).
Next three terms: $22 \div 2 = 11$, $11 \div 2 = \frac{11}{2}$, $\frac{11}{2} \div 2 = \frac{11}{4}$ → $\mathbf{11, \frac{11}{2}, \frac{11}{4}}$.
(ix) $3200, 32, 0.32, \dots$:
Rule: Divide by $100$ (multiply by $0.01$).
Next three terms: $\mathbf{0.0032, 0.000032, 0.00000032}$.
Rule: Divide by $100$ (multiply by $0.01$).
Next three terms: $\mathbf{0.0032, 0.000032, 0.00000032}$.
(x) $100, 89, 78, \dots$:
Rule: Subtract $11$ from previous term.
Next three terms: $\mathbf{67, 56, 45}$.
Rule: Subtract $11$ from previous term.
Next three terms: $\mathbf{67, 56, 45}$.
Question 2
Exercise 6.7
Identify the triangular patterns and complete the table below:
| Number of Triangles ($T$) | 1 | 2 | 3 | 4 | 5 |
| Number of Lines ($L$) | 3 | 5 | 7 | 9 | 11 |
(a) Write the rule in words: The number of lines is equal to two times the number of triangles plus one.
(b) Write the rule as a formula: $\mathbf{L = 2T + 1}$.
(c) Find number of lines required to construct 100 triangles:
Substitute $T = 100$ into formula: $L = 2(100) + 1 = 200 + 1 = \mathbf{201}$ lines.
Substitute $T = 100$ into formula: $L = 2(100) + 1 = 200 + 1 = \mathbf{201}$ lines.
Questions 3 & 4: Sequence Construction
Exercise 6.7
Question 3: Create sequences with specified rules (e.g. starting values)
(i) Add 3 to each term (start with 2):
$2, 5, 8, 11, 14, \dots$
$2, 5, 8, 11, 14, \dots$
(ii) Multiply by $\frac{1}{3}$ (start with 27):
$27, 9, 3, 1, \frac{1}{3}, \dots$
$27, 9, 3, 1, \frac{1}{3}, \dots$
(iii) Subtract 2 from each term (start with 20):
$20, 18, 16, 14, 12, \dots$
$20, 18, 16, 14, 12, \dots$
(iv) Divide each term by 2 (start with 32):
$32, 16, 8, 4, 2, \dots$
$32, 16, 8, 4, 2, \dots$
(v) Add 0.1, 0.2, 0.3... (start with 1):
$1, 1.1, 1.3, 1.6, 2.0, \dots$
$1, 1.1, 1.3, 1.6, 2.0, \dots$
Question 4: Sequence generation with fixed starting numbers
(i) Add 0.6 to each term; start with 10:
$\mathbf{10, 10.6, 11.2, 11.8, 12.4, \dots}$
$\mathbf{10, 10.6, 11.2, 11.8, 12.4, \dots}$
(ii) Square consecutive odd integers; start with 1:
$1^2, 3^2, 5^2, 7^2 → \mathbf{1, 9, 25, 49, \dots}$
$1^2, 3^2, 5^2, 7^2 → \mathbf{1, 9, 25, 49, \dots}$
(iii) Add $\frac{1}{2}$ times each term; start with 12:
$12, 12+6=18, 18+9=27, 27+13.5=40.5 → \mathbf{12, 18, 27, 40.5, \dots}$
$12, 12+6=18, 18+9=27, 27+13.5=40.5 → \mathbf{12, 18, 27, 40.5, \dots}$
(iv) Cube consecutive even integers; start with 2:
$2^3, 4^3, 6^3, 8^3 → \mathbf{8, 64, 216, 512, \dots}$
$2^3, 4^3, 6^3, 8^3 → \mathbf{8, 64, 216, 512, \dots}$
(v) Divide each term by 4; start with 4:
$\mathbf{4, 1, \frac{1}{4}, \frac{1}{16}, \dots}$
$\mathbf{4, 1, \frac{1}{4}, \frac{1}{16}, \dots}$
Questions 5 & 6: General Term Evaluation
Exercise 6.7
Question 5: First five terms of sequence:
• (i) General term $2n + 1$:
$n=1: 2(1)+1 = 3$
$n=2: 2(2)+1 = 5$
$n=3: 2(3)+1 = 7$
$n=4: 2(4)+1 = 9$
$n=5: 2(5)+1 = 11$
First five terms: $\mathbf{3, 5, 7, 9, 11}$.
• (ii) General term $x^2 - 1$:
$x=1: 1^2 - 1 = 0$
$x=2: 2^2 - 1 = 3$
$x=3: 3^2 - 1 = 8$
$x=4: 4^2 - 1 = 15$
$x=5: 5^2 - 1 = 24$
First five terms: $\mathbf{0, 3, 8, 15, 24}$.
• (i) General term $2n + 1$:
$n=1: 2(1)+1 = 3$
$n=2: 2(2)+1 = 5$
$n=3: 2(3)+1 = 7$
$n=4: 2(4)+1 = 9$
$n=5: 2(5)+1 = 11$
First five terms: $\mathbf{3, 5, 7, 9, 11}$.
• (ii) General term $x^2 - 1$:
$x=1: 1^2 - 1 = 0$
$x=2: 2^2 - 1 = 3$
$x=3: 3^2 - 1 = 8$
$x=4: 4^2 - 1 = 15$
$x=5: 5^2 - 1 = 24$
First five terms: $\mathbf{0, 3, 8, 15, 24}$.
Question 6: Given general term $3y + 4$, find:
• (i) 7th term ($y = 7$):
$T_7 = 3(7) + 4 = 21 + 4 = \mathbf{25}$.
• (ii) 10th term ($y = 10$):
$T_{10} = 3(10) + 4 = 30 + 4 = \mathbf{34}$.
• (i) 7th term ($y = 7$):
$T_7 = 3(7) + 4 = 21 + 4 = \mathbf{25}$.
• (ii) 10th term ($y = 10$):
$T_{10} = 3(10) + 4 = 30 + 4 = \mathbf{34}$.
7. Review Exercise 6 • 100% Complete Step-by-Step Solved Solutions
Question 1: Multiple Choice Questions (MCQs)
Review Exercise 6
(i) $8$ is a polynomial of degree:
(a) $0$ (b) $1$ (c) $2$ (d) $4$
Correct Answer: (a) 0 — Any non-zero constant $c$ can be written as $c \cdot x^0$, which has degree $0$.
(a) $0$ (b) $1$ (c) $2$ (d) $4$
Correct Answer: (a) 0 — Any non-zero constant $c$ can be written as $c \cdot x^0$, which has degree $0$.
(ii) $(x^2 + 3)(x^2 - 6)$ is a polynomial of degree:
(a) $4$ (b) $2$ (c) $3$ (d) $1$
Correct Answer: (a) 4 — Highest power term is $x^2 \times x^2 = x^4$, so degree is $4$.
(a) $4$ (b) $2$ (c) $3$ (d) $1$
Correct Answer: (a) 4 — Highest power term is $x^2 \times x^2 = x^4$, so degree is $4$.
(iii) $0$ is a polynomial of:
(a) degree 1 (b) degree 2 (c) degree 0 (d) No degree
Correct Answer: (c) degree 0 — Following the textbook answer key convention (page 278). Note: In advanced mathematics, the zero polynomial has an undefined degree.
(a) degree 1 (b) degree 2 (c) degree 0 (d) No degree
Correct Answer: (c) degree 0 — Following the textbook answer key convention (page 278). Note: In advanced mathematics, the zero polynomial has an undefined degree.
(iv) $3x^3 + 2x^2 + 1$ is polynomial of degree:
(a) $2$ (b) $3$ (c) $1$ (d) $4$
Correct Answer: (b) 3 — Highest exponent of $x$ is $3$.
(a) $2$ (b) $3$ (c) $1$ (d) $4$
Correct Answer: (b) 3 — Highest exponent of $x$ is $3$.
(v) The product of $(\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b})$ is:
(a) $a + b$ (b) $a^2 - b^2$ (c) $a^2 + b^2$ (d) $a - b$
Correct Answer: (d) $a - b$ — By difference of squares: $(\sqrt{a})^2 - (\sqrt{b})^2 = a - b$.
(a) $a + b$ (b) $a^2 - b^2$ (c) $a^2 + b^2$ (d) $a - b$
Correct Answer: (d) $a - b$ — By difference of squares: $(\sqrt{a})^2 - (\sqrt{b})^2 = a - b$.
(vi) The square of $x + 2y$ is:
(a) $x^2 + 4y^2$ (b) $x^2 + 4y^2 + 4xy$ (c) $x^2 + 4y^2 - 4xy$ (d) $x^2 - 4y^2$
Correct Answer: (b) $x^2 + 4y^2 + 4xy$ — $(x + 2y)^2 = x^2 + 2(x)(2y) + (2y)^2 = x^2 + 4xy + 4y^2$.
(a) $x^2 + 4y^2$ (b) $x^2 + 4y^2 + 4xy$ (c) $x^2 + 4y^2 - 4xy$ (d) $x^2 - 4y^2$
Correct Answer: (b) $x^2 + 4y^2 + 4xy$ — $(x + 2y)^2 = x^2 + 2(x)(2y) + (2y)^2 = x^2 + 4xy + 4y^2$.
(vii) The square of $y - 2x$ is:
(a) $y^2 + 4x^2 - 4xy$ (b) $y^2 + 4x^2 + 4xy$ (c) $y^2 + 4x^2$ (d) $y^2 - 4x^2$
Correct Answer: (a) $y^2 + 4x^2 - 4xy$ — $(y - 2x)^2 = y^2 - 2(y)(2x) + (2x)^2 = y^2 - 4xy + 4x^2$.
(a) $y^2 + 4x^2 - 4xy$ (b) $y^2 + 4x^2 + 4xy$ (c) $y^2 + 4x^2$ (d) $y^2 - 4x^2$
Correct Answer: (a) $y^2 + 4x^2 - 4xy$ — $(y - 2x)^2 = y^2 - 2(y)(2x) + (2x)^2 = y^2 - 4xy + 4x^2$.
(viii) The product of $2x + 3$ and $x$ is:
(a) $2x^2 + 3$ (b) $2x^2 - 3x$ (c) $2x^2 + 3x$ (d) $2x^2 + 3x + 6$
Correct Answer: (c) $2x^2 + 3x$ — $x(2x + 3) = 2x^2 + 3x$.
(a) $2x^2 + 3$ (b) $2x^2 - 3x$ (c) $2x^2 + 3x$ (d) $2x^2 + 3x + 6$
Correct Answer: (c) $2x^2 + 3x$ — $x(2x + 3) = 2x^2 + 3x$.
(ix) The simplified form of $(a + b) - (a - b)$ is:
(a) $4ab$ (b) $2a + 2b$ (c) $2b$ (d) $2a$
Correct Answer: (c) $2b$ — $(a + b) - (a - b) = a + b - a + b = 2b$.
(a) $4ab$ (b) $2a + 2b$ (c) $2b$ (d) $2a$
Correct Answer: (c) $2b$ — $(a + b) - (a - b) = a + b - a + b = 2b$.
(x) $3, 1, \frac{1}{3}, \frac{1}{9}, \dots$ the next term of the sequence is:
(a) $\frac{1}{21}$ (b) $\frac{1}{27}$ (c) $\frac{1}{81}$ (d) $\frac{1}{243}$
Correct Answer: (b) $\frac{1}{27}$ — Each term is multiplied by $\frac{1}{3}$: $\frac{1}{9} \times \frac{1}{3} = \frac{1}{27}$.
(a) $\frac{1}{21}$ (b) $\frac{1}{27}$ (c) $\frac{1}{81}$ (d) $\frac{1}{243}$
Correct Answer: (b) $\frac{1}{27}$ — Each term is multiplied by $\frac{1}{3}$: $\frac{1}{9} \times \frac{1}{3} = \frac{1}{27}$.
Questions 2 to 6: Geometric & Algebraic Simplifications
Review Exercise 6
Question 2: Find the perimeter of a square of side $x + y$.
Solution:
Perimeter of square $= 4 \times \text{side} = 4(x + y) = \mathbf{4x + 4y}$ (or $4(x + y)$).
Solution:
Perimeter of square $= 4 \times \text{side} = 4(x + y) = \mathbf{4x + 4y}$ (or $4(x + y)$).
Question 3: Find the area of a square of side $4a - b$.
Solution:
Area of square $= (\text{side})^2 = (4a - b)^2 = (4a)^2 - 2(4a)(b) + b^2 = \mathbf{16a^2 - 8ab + b^2}$.
Solution:
Area of square $= (\text{side})^2 = (4a - b)^2 = (4a)^2 - 2(4a)(b) + b^2 = \mathbf{16a^2 - 8ab + b^2}$.
Question 4: Find the area and perimeter of the rectangle with dimensions $5x + 1$ and $5x - 1$.
Solution:
• Area: $(5x + 1)(5x - 1) = (5x)^2 - 1^2 = \mathbf{25x^2 - 1}$.
• Perimeter: $2((5x + 1) + (5x - 1)) = 2(10x) = \mathbf{20x}$.
Solution:
• Area: $(5x + 1)(5x - 1) = (5x)^2 - 1^2 = \mathbf{25x^2 - 1}$.
• Perimeter: $2((5x + 1) + (5x - 1)) = 2(10x) = \mathbf{20x}$.
Question 5: Simplify $(2x + 1)(2x - 1) - (2x - 1)^2$.
Solution:
First term: $(2x + 1)(2x - 1) = 4x^2 - 1$.
Second term: $(2x - 1)^2 = 4x^2 - 4x + 1$.
Difference: $(4x^2 - 1) - (4x^2 - 4x + 1) = 4x^2 - 1 - 4x^2 + 4x - 1 = \mathbf{4x - 2}$.
Solution:
First term: $(2x + 1)(2x - 1) = 4x^2 - 1$.
Second term: $(2x - 1)^2 = 4x^2 - 4x + 1$.
Difference: $(4x^2 - 1) - (4x^2 - 4x + 1) = 4x^2 - 1 - 4x^2 + 4x - 1 = \mathbf{4x - 2}$.
Question 6: Simplify $60(a + b)^2 - 45(a - b)^2$.
Solution:
$= 60(a^2 + 2ab + b^2) - 45(a^2 - 2ab + b^2)$
$= 60a^2 + 120ab + 60b^2 - 45a^2 + 90ab - 45b^2$
$= 15a^2 + 210ab + 15b^2 = \mathbf{15(a^2 + 14ab + b^2)}$.
Solution:
$= 60(a^2 + 2ab + b^2) - 45(a^2 - 2ab + b^2)$
$= 60a^2 + 120ab + 60b^2 - 45a^2 + 90ab - 45b^2$
$= 15a^2 + 210ab + 15b^2 = \mathbf{15(a^2 + 14ab + b^2)}$.
Questions 7 to 11: Expressions, Sequences & Financial Applications
Review Exercise 6
Question 7: Simplify $(y - z)^2 - (y + z)^2$.
Solution:
$= (y^2 - 2yz + z^2) - (y^2 + 2yz + z^2) = y^2 - 2yz + z^2 - y^2 - 2yz - z^2 = \mathbf{-4yz}$.
Solution:
$= (y^2 - 2yz + z^2) - (y^2 + 2yz + z^2) = y^2 - 2yz + z^2 - y^2 - 2yz - z^2 = \mathbf{-4yz}$.
Question 8: Select a number of your choice and create a sequence by multiplying each term by $\frac{1}{2}$.
Solution:
Let starting number be $64$.
Sequence: $64, 64 \times \frac{1}{2} = 32, 32 \times \frac{1}{2} = 16, 16 \times \frac{1}{2} = 8, 4, 2, 1, \frac{1}{2}, \dots$
Final Answer: $\mathbf{64, 32, 16, 8, 4, 2, \dots}$
Solution:
Let starting number be $64$.
Sequence: $64, 64 \times \frac{1}{2} = 32, 32 \times \frac{1}{2} = 16, 16 \times \frac{1}{2} = 8, 4, 2, 1, \frac{1}{2}, \dots$
Final Answer: $\mathbf{64, 32, 16, 8, 4, 2, \dots}$
Question 9: Given the sequence $2, 5, 8, 11, \dots$
(i) Find its general term.
(ii) Find 100th term using general term.
Solution:
First term $a = 2$. Common difference $d = 5 - 2 = 3$.
• (i) General term: $T_n = a + (n - 1)d = 2 + (n - 1)3 = 2 + 3n - 3 = \mathbf{3n - 1}$.
• (ii) 100th term: $T_{100} = 3(100) - 1 = 300 - 1 = \mathbf{299}$.
(i) Find its general term.
(ii) Find 100th term using general term.
Solution:
First term $a = 2$. Common difference $d = 5 - 2 = 3$.
• (i) General term: $T_n = a + (n - 1)d = 2 + (n - 1)3 = 2 + 3n - 3 = \mathbf{3n - 1}$.
• (ii) 100th term: $T_{100} = 3(100) - 1 = 300 - 1 = \mathbf{299}$.
Question 10: Find the 7th term of sequence $1, 3, 7, 15, \dots$
Solution:
Examine the sequence: $1 = 2^1 - 1$, $3 = 2^2 - 1$, $7 = 2^3 - 1$, $15 = 2^4 - 1$.
The general formula is $T_n = 2^n - 1$.
For the 7th term ($n = 7$): $T_7 = 2^7 - 1 = 128 - 1 = \mathbf{127}$.
Solution:
Examine the sequence: $1 = 2^1 - 1$, $3 = 2^2 - 1$, $7 = 2^3 - 1$, $15 = 2^4 - 1$.
The general formula is $T_n = 2^n - 1$.
For the 7th term ($n = 7$): $T_7 = 2^7 - 1 = 128 - 1 = \mathbf{127}$.
Question 11: Abid deposited Rs. 1000 in the bank in January. Next month, he deposited Rs. 1250 and continued to deposit in the same ratio each month (adding Rs. 250 each month).
(i) Find the amount deposited in the month of August.
(ii) Find total amount deposited in his account.
Solution:
January is month $n = 1$, February is $n = 2$, August is month $n = 8$.
First term $a = 1000$, monthly increase $d = 250$.
• Part (i): Amount deposited in August ($n = 8$):
$T_8 = a + (8 - 1)d = 1000 + 7(250) = 1000 + 1750 = \mathbf{\text{Rs. } 2750}$.
• Part (ii): Total amount deposited:
Sum of arithmetic progression for 8 months ($n = 8$):
$S_8 = \frac{n}{2}[a + T_8] = \frac{8}{2}[1000 + 2750] = 4 \times 3750 = \mathbf{\text{Rs. } 15000}$.
Note on Textbook Answer Key: The textbook answer key (page 278) records Rs. 14,100, which reflects a minor clerical tally error in the published print. The exact mathematical sum is Rs. 15,000.
(i) Find the amount deposited in the month of August.
(ii) Find total amount deposited in his account.
Solution:
January is month $n = 1$, February is $n = 2$, August is month $n = 8$.
First term $a = 1000$, monthly increase $d = 250$.
• Part (i): Amount deposited in August ($n = 8$):
$T_8 = a + (8 - 1)d = 1000 + 7(250) = 1000 + 1750 = \mathbf{\text{Rs. } 2750}$.
• Part (ii): Total amount deposited:
Sum of arithmetic progression for 8 months ($n = 8$):
$S_8 = \frac{n}{2}[a + T_8] = \frac{8}{2}[1000 + 2750] = 4 \times 3750 = \mathbf{\text{Rs. } 15000}$.
Note on Textbook Answer Key: The textbook answer key (page 278) records Rs. 14,100, which reflects a minor clerical tally error in the published print. The exact mathematical sum is Rs. 15,000.
8. Unit 06 Algebra • Complete Formula Summary & Active Recall
Essential Algebraic Identities & Rules Cheat Sheet
| Concept / Identity | Mathematical Formula | Application / Purpose |
|---|---|---|
| Identity I (Square of Sum) | $(a + b)^2 = a^2 + 2ab + b^2$ | Expanding sums, rapid squares like $102^2 = (100+2)^2$ |
| Identity II (Square of Diff) | $(a - b)^2 = a^2 - 2ab + b^2$ | Expanding differences, rapid squares like $999^2 = (1000-1)^2$ |
| Identity III (Cross Product) | $(x + a)(x + b) = x^2 + (a + b)x + ab$ | Mental products like $103 \times 96 = (100+3)(100-4)$ |
| Identity IV (Diff of Squares) | $(a + b)(a - b) = a^2 - b^2$ | Mental products like $52 \times 48 = 50^2 - 2^2 = 2496$ |
| Continuous Products | $(a - b)(a + b)(a^2 + b^2) = a^4 - b^4$ | Repeated application of difference of two squares |
| Common Factoring | $ka + kb + kc = k(a + b + c)$ | Extracting GCF of coefficients and variables |
| Grouping Method | $ac + ad + bc + bd = (a + b)(c + d)$ | Factoring 4-term expressions in pairs |
| Middle Term Splitting | $ax^2 + bx + c$, find $p+q=b, pq=ac$ | Factoring quadratic trinomials into linear binomials |
| Arithmetic General Term | $T_n = a + (n - 1)d$ | Finding the $n$-th term of any constant-difference sequence |
| Fibonacci Sequence | $1, 1, 2, 3, 5, 8, 13, 21, 34, 55, \dots$ | Natural spiral patterns in sunflowers, pine cones, shell growth |
Active Recall & Self-Test Flashcards
Test your understanding before practice tests! Click on any question to reveal the verified answer.
1. Why is an expression with $\\frac{1}{x}$ not a polynomial?
Because $\\frac{1}{x} = x^{-1}$, which has a negative exponent ($-1$). A polynomial strictly requires all variable exponents to be non-negative whole numbers ($0, 1, 2, 3, \dots$).
2. What is the degree and leading coefficient of $7 - 2x + 9x^4$?
Arrange in descending order: $9x^4 - 2x + 7$. The highest power is $4$, so degree $= 4$. The coefficient of this highest term is leading coefficient $= 9$.
3. Why is $(a + b)^2$ NOT equal to $a^2 + b^2$?
By distributive multiplication, $(a + b)(a + b) = a(a + b) + b(a + b) = a^2 + ab + ba + b^2 = a^2 + 2ab + b^2$. Omitting the middle cross-term $2ab$ loses the area of the two side rectangles!
4. How do you mentally compute $98 \times 102$ in 2 seconds?
Express as difference of squares: $(100 - 2)(100 + 2) = 100^2 - 2^2 = 10,000 - 4 = \mathbf{9996}$.
5. In the arithmetic sequence $5, 9, 13, 17, \dots$, what is the 50th term?
First term $a = 5$, common difference $d = 9 - 5 = 4$.
$T_{50} = a + (50 - 1)d = 5 + 49(4) = 5 + 196 = \mathbf{201}$.
$T_{50} = a + (50 - 1)d = 5 + 49(4) = 5 + 196 = \mathbf{201}$.
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