Class 8
Mathematics
Federal Board of Intermediate and Secondary Education (FBISE) (FBISE)
📚 Model Textbook of Mathematics Grade 8 (FBISE / NBF)
Class 8 Mathematics - Ch 6: Algebra & Sequence Mastery Guide (FBISE)
📖 Chapter 6: Algebra and Sequence
📅 Updated: Sep 14, 2026
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Teacher & Parent Instructional Roadmap
Unit Focus: Transitioning middle-school students from basic arithmetic into abstract algebraic reasoning, structural polynomial manipulations, exponent laws, binomial expansions, multi-term factorizations, and systematic pattern modeling via Arithmetic and Geometric sequences.
🎯 Student Learning Outcomes (SLOs):
- Perform addition, subtraction, and multi-term multiplication of polynomials (FOIL method).
- Divide degree-3 polynomials by monomials, linear binomials, and quadratics using long division.
- Master algebraic identities: $(a \pm b)^2 = a^2 \pm 2ab + b^2$, $(a+b)(a-b) = a^2 - b^2$, and $(a \pm b)^3$.
- Factorize trinomials and 4-term expressions of the form $a^2 \pm 2ab + b^2 - c^2$.
- Differentiate between Arithmetic Progression ($a_n = a_1 + (n-1)d$) and Geometric Progression ($a_n = a_1 r^{n-1}$).
⚠️ Common Student Pitfalls & Misconceptions:
- The "Freshman's Dream" Error: Writing $(a+b)^2 = a^2 + b^2$ while forgetting the middle cross-term $+2ab$.
- Sign Distribution: Forgetting to distribute negative signs across inner brackets in subtraction: $-(a - b) = -a + b$.
- Exponent Multiplication vs Addition: Confusing $x^m \cdot x^n = x^{m+n}$ with $(x^m)^n = x^{mn}$.
- AP vs GP Difference: Subtracting consecutive terms for AP ($d = a_2 - a_1$) vs dividing consecutive terms for GP ($r = a_2 / a_1$).
⚡ The 3-Step Algebraic Mastery Strategy:
1. Identify Structure: Check if the expression is a monomial, binomial, perfect square, difference of squares, or arithmetic/geometric series.
2. Apply Exact Law / Formula: Match standard identities: $(a \pm b)^2$, $(a^2 - b^2)$, $(a \pm b)^3$, or $a_n = a_1 + (n-1)d$.
3. Verify by Expansion / Substitution: Multiply factors back or substitute $n=1, 2$ to verify that LHS identically equals RHS.
2. Apply Exact Law / Formula: Match standard identities: $(a \pm b)^2$, $(a^2 - b^2)$, $(a \pm b)^3$, or $a_n = a_1 + (n-1)d$.
3. Verify by Expansion / Substitution: Multiply factors back or substitute $n=1, 2$ to verify that LHS identically equals RHS.
🌍 Real-World Connections
Algebra and Sequences power modern science, engineering, and economics:
• Architectural Areas: Calculating floor space and perimeter borders when room dimensions change dynamically.
• Financial Salary Growth: Career salary ladders and annual increments follow Arithmetic Progressions (AP).
• Viral Population Growth: Bacteria cell division and viral social media shares double geometrically ($2^n$ in GP).
💡 Kid-Friendly Memory Hooks
• FOIL for Binomials: Firsts, Outsides, Insides, Lasts!
• Zero Exponent Rule: Any non-zero base raised to power zero equals $1$ ($x^0 = 1$), because dividing anything by itself is 1 ($\frac{x^n}{x^n} = x^{n-n} = x^0 = 1$).
• Difference of Two Squares: $a^2 - b^2 = (a+b)(a-b)$ — "One plus, one minus, clean and fast!"
Visual Concept 1: Geometric Proof of $(a+b)^2 = a^2 + 2ab + b^2$ and FOIL Method
Visual Concept 2: Laws of Exponents & Polynomial Long Division Flow
Visual Concept 3: Arithmetic Sequence (AP) vs Geometric Sequence (GP)
Section-by-Section Theoretical Mastery
1. Polynomials: Classification by Terms & Degree
A polynomial is an algebraic expression consisting of variables with non-negative integer exponents, real coefficients, and arithmetic operations:
Monomial (1 Term): $5x^3, -7, 4ab$
Binomial (2 Terms): $2x + 5, x^2 - y^2$
Trinomial (3 Terms): $ax^2 + bx + c$
Degree: Highest power of variable in the expression.
2. Laws of Exponents Summary
1. Product Law: $x^m \times x^n = x^{m+n}$ and $(xy)^n = x^n y^n$
2. Quotient Law: $x^m \div x^n = x^{m-n}$ and $(x/y)^n = x^n / y^n$ ($x, y \neq 0$)
3. Power of a Power: $(x^m)^n = x^{mn}$
4. Zero Exponent: $x^0 = 1$ ($x \neq 0$)
5. Negative Exponent: $x^{-n} = \frac{1}{x^n}$ and $\frac{1}{x^{-n}} = x^n$
2. Quotient Law: $x^m \div x^n = x^{m-n}$ and $(x/y)^n = x^n / y^n$ ($x, y \neq 0$)
3. Power of a Power: $(x^m)^n = x^{mn}$
4. Zero Exponent: $x^0 = 1$ ($x \neq 0$)
5. Negative Exponent: $x^{-n} = \frac{1}{x^n}$ and $\frac{1}{x^{-n}} = x^n$
3. Core Factorization Types
• Common Monomial: $ka + kb + kc = k(a + b + c)$
• Perfect Square Trinomial: $a^2 \pm 2ab + b^2 = (a \pm b)^2$
• Difference of Two Squares: $a^2 - b^2 = (a + b)(a - b)$
• Four-Term Expressions: $a^2 \pm 2ab + b^2 - c^2 = (a \pm b)^2 - c^2 = (a \pm b + c)(a \pm b - c)$
• Cube of Binomials: $(a \pm b)^3 = a^3 \pm 3a^2b + 3ab^2 \pm b^3 = a^3 \pm 3ab(a \pm b) \pm b^3$
• Perfect Square Trinomial: $a^2 \pm 2ab + b^2 = (a \pm b)^2$
• Difference of Two Squares: $a^2 - b^2 = (a + b)(a - b)$
• Four-Term Expressions: $a^2 \pm 2ab + b^2 - c^2 = (a \pm b)^2 - c^2 = (a \pm b + c)(a \pm b - c)$
• Cube of Binomials: $(a \pm b)^3 = a^3 \pm 3a^2b + 3ab^2 \pm b^3 = a^3 \pm 3ab(a \pm b) \pm b^3$
Exercise 6.1 • Polynomial Addition, Subtraction & FOIL Multiplication
Q1. Find the sum of the polynomials:
(i) $2x^3 - 5x^2 + x$ and $2x^2 + x^3 - 1$:
$$\text{Sum} = (2x^3 + x^3) + (-5x^2 + 2x^2) + x - 1 = \mathbf{3x^3 - 3x^2 + x - 1}$$
(ii) $3x^2 + 2x + 1$ and $x^2 - x - 1$:
$$\text{Sum} = (3x^2 + x^2) + (2x - x) + (1 - 1) = \mathbf{4x^2 + x}$$
(iii) $x^3 + 2x^2 - 1$ and $2x^2 + x^3$:
$$\text{Sum} = (x^3 + x^3) + (2x^2 + 2x^2) - 1 = \mathbf{2x^3 + 4x^2 - 1}$$
(iv) $4y^2 + 5$ and $-2y^2 + 2y - 4$:
$$\text{Sum} = (4y^2 - 2y^2) + 2y + (5 - 4) = \mathbf{2y^2 + 2y + 1}$$
(v) $3a^2 + 2a - 5$ and $-a^2 - a + 3$:
$$\text{Sum} = (3a^2 - a^2) + (2a - a) + (-5 + 3) = \mathbf{2a^2 + a - 2}$$
(vi) $x^3 + x^2 + x + 1$ and $x^3 - x^2 + x - 1$:
$$\text{Sum} = (x^3 + x^3) + (x^2 - x^2) + (x + x) + (1 - 1) = \mathbf{2x^3 + 2x}$$
$$\text{Sum} = (2x^3 + x^3) + (-5x^2 + 2x^2) + x - 1 = \mathbf{3x^3 - 3x^2 + x - 1}$$
(ii) $3x^2 + 2x + 1$ and $x^2 - x - 1$:
$$\text{Sum} = (3x^2 + x^2) + (2x - x) + (1 - 1) = \mathbf{4x^2 + x}$$
(iii) $x^3 + 2x^2 - 1$ and $2x^2 + x^3$:
$$\text{Sum} = (x^3 + x^3) + (2x^2 + 2x^2) - 1 = \mathbf{2x^3 + 4x^2 - 1}$$
(iv) $4y^2 + 5$ and $-2y^2 + 2y - 4$:
$$\text{Sum} = (4y^2 - 2y^2) + 2y + (5 - 4) = \mathbf{2y^2 + 2y + 1}$$
(v) $3a^2 + 2a - 5$ and $-a^2 - a + 3$:
$$\text{Sum} = (3a^2 - a^2) + (2a - a) + (-5 + 3) = \mathbf{2a^2 + a - 2}$$
(vi) $x^3 + x^2 + x + 1$ and $x^3 - x^2 + x - 1$:
$$\text{Sum} = (x^3 + x^3) + (x^2 - x^2) + (x + x) + (1 - 1) = \mathbf{2x^3 + 2x}$$
Q2. Subtract the second polynomial from the first:
(i) $(4x^2 + 2x - 3) - (2x^2 - x + 1)$:
$$= 4x^2 + 2x - 3 - 2x^2 + x - 1 = \mathbf{2x^2 + 3x - 4}$$
(ii) $(3y^3 - 2y^2 + y - 4) - (y^3 + 3y^2 - 2y + 5)$:
$$= (3y^3 - y^3) + (-2y^2 - 3y^2) + (y + 2y) + (-4 - 5) = \mathbf{2y^3 - 5y^2 + 3y - 9}$$
(iii) $(5a^2 - 3a + 7) - (-2a^2 + 4a - 1)$:
$$= 5a^2 - 3a + 7 + 2a^2 - 4a + 1 = \mathbf{7a^2 - 7a + 8}$$
(iv) $(x^3 - 1) - (x^2 + x + 1)$:
$$= \mathbf{x^3 - x^2 - x - 2}$$
$$= 4x^2 + 2x - 3 - 2x^2 + x - 1 = \mathbf{2x^2 + 3x - 4}$$
(ii) $(3y^3 - 2y^2 + y - 4) - (y^3 + 3y^2 - 2y + 5)$:
$$= (3y^3 - y^3) + (-2y^2 - 3y^2) + (y + 2y) + (-4 - 5) = \mathbf{2y^3 - 5y^2 + 3y - 9}$$
(iii) $(5a^2 - 3a + 7) - (-2a^2 + 4a - 1)$:
$$= 5a^2 - 3a + 7 + 2a^2 - 4a + 1 = \mathbf{7a^2 - 7a + 8}$$
(iv) $(x^3 - 1) - (x^2 + x + 1)$:
$$= \mathbf{x^3 - x^2 - x - 2}$$
Q3 & Q4. Multiply the polynomials (FOIL & Distributive Laws):
(i) $(x + 3)(x + 5)$: $= x^2 + 5x + 3x + 15 = \mathbf{x^2 + 8x + 15}$
(ii) $(2x - 1)(x + 4)$: $= 2x^2 + 8x - x - 4 = \mathbf{2x^2 + 7x - 4}$
(iii) $(3a + 2)(2a - 3)$: $= 6a^2 - 9a + 4a - 6 = \mathbf{6a^2 - 5a - 6}$
(iv) $(x^2 + 2x + 1)(x - 1)$: $= x(x^2 + 2x + 1) - 1(x^2 + 2x + 1) = x^3 + 2x^2 + x - x^2 - 2x - 1 = \mathbf{x^3 + x^2 - x - 1}$
(v) $(y - 3)(y^2 + 3y + 9)$: $= y^3 + 3y^2 + 9y - 3y^2 - 9y - 27 = \mathbf{y^3 - 27}$
(ii) $(2x - 1)(x + 4)$: $= 2x^2 + 8x - x - 4 = \mathbf{2x^2 + 7x - 4}$
(iii) $(3a + 2)(2a - 3)$: $= 6a^2 - 9a + 4a - 6 = \mathbf{6a^2 - 5a - 6}$
(iv) $(x^2 + 2x + 1)(x - 1)$: $= x(x^2 + 2x + 1) - 1(x^2 + 2x + 1) = x^3 + 2x^2 + x - x^2 - 2x - 1 = \mathbf{x^3 + x^2 - x - 1}$
(v) $(y - 3)(y^2 + 3y + 9)$: $= y^3 + 3y^2 + 9y - 3y^2 - 9y - 27 = \mathbf{y^3 - 27}$
Q5. Write a polynomial representing the area of the shaded region:
(i) Rectangle with width $(2x - 5)$ and height $(x + 5)$:
$$\text{Area} = (2x - 5)(x + 5) = 2x^2 + 10x - 5x - 25 = \mathbf{2x^2 + 5x - 25}$$
(ii) Sub-rectangle with dimensions $x \times 2x$:
$$\text{Area} = x(2x) = \mathbf{2x^2}$$
(iii) Shaded sub-square of dimensions $x \times x$:
$$\text{Area} = x \times x = \mathbf{x^2}$$
(iv) Outer rectangle $10 \times 8$ minus inner unshaded $3x \times 2x$:
$$\text{Area} = (10 \times 8) - (3x \times 2x) = \mathbf{80 - 6x^2}$$
(v) Outer square $(x + 1)(x + 1)$ minus inner rectangle $5(x - 7)$:
$$\text{Area} = (x^2 + 2x + 1) - (5x - 35) = x^2 + 2x + 1 - 5x + 35 = \mathbf{x^2 - 3x + 36}$$
$$\text{Area} = (2x - 5)(x + 5) = 2x^2 + 10x - 5x - 25 = \mathbf{2x^2 + 5x - 25}$$
(ii) Sub-rectangle with dimensions $x \times 2x$:
$$\text{Area} = x(2x) = \mathbf{2x^2}$$
(iii) Shaded sub-square of dimensions $x \times x$:
$$\text{Area} = x \times x = \mathbf{x^2}$$
(iv) Outer rectangle $10 \times 8$ minus inner unshaded $3x \times 2x$:
$$\text{Area} = (10 \times 8) - (3x \times 2x) = \mathbf{80 - 6x^2}$$
(v) Outer square $(x + 1)(x + 1)$ minus inner rectangle $5(x - 7)$:
$$\text{Area} = (x^2 + 2x + 1) - (5x - 35) = x^2 + 2x + 1 - 5x + 35 = \mathbf{x^2 - 3x + 36}$$
Exercise 6.2 • Laws of Exponents & Simplification
Q1. Simplify by using Product and Quotient Laws:
(i) $3^5 \times 3^8 = 3^{5+8} = \mathbf{3^{13}}$
(ii) $4^8 \times 4^5 = 4^{8+5} = \mathbf{4^{13}}$
(iii) $\left(-\frac{9}{11}\right)^7 \times \left(-\frac{9}{11}\right)^3 = \left(-\frac{9}{11}\right)^{7+3} = \mathbf{\left(-\frac{9}{11}\right)^{10}}$
(iv) $(-2)^6 \div (-2)^3 = (-2)^{6-3} = \mathbf{(-2)^3 = -8}$
(v) $6^3 \div 5^3 = \mathbf{\left(\frac{6}{5}\right)^3}$
(vi) $\left(-\frac{2}{3}\right)^7 \div \left(-\frac{2}{3}\right)^5 = \left(-\frac{2}{3}\right)^{7-5} = \mathbf{\left(-\frac{2}{3}\right)^2 = \frac{4}{9}}$
(ii) $4^8 \times 4^5 = 4^{8+5} = \mathbf{4^{13}}$
(iii) $\left(-\frac{9}{11}\right)^7 \times \left(-\frac{9}{11}\right)^3 = \left(-\frac{9}{11}\right)^{7+3} = \mathbf{\left(-\frac{9}{11}\right)^{10}}$
(iv) $(-2)^6 \div (-2)^3 = (-2)^{6-3} = \mathbf{(-2)^3 = -8}$
(v) $6^3 \div 5^3 = \mathbf{\left(\frac{6}{5}\right)^3}$
(vi) $\left(-\frac{2}{3}\right)^7 \div \left(-\frac{2}{3}\right)^5 = \left(-\frac{2}{3}\right)^{7-5} = \mathbf{\left(-\frac{2}{3}\right)^2 = \frac{4}{9}}$
Q2, Q3 & Q4. Power Law, Zero & Negative Exponents:
(i) $(2^3)^4 = 2^{3 \times 4} = \mathbf{2^{12} = 4096}$
(ii) $(x^5)^3 = x^{5 \times 3} = \mathbf{x^{15}}$
(iii) $(a^{-2})^{-4} = a^{(-2)(-4)} = \mathbf{a^8}$
(iv) $\left(\frac{x^2 y^3}{z^4}\right)^2 = \mathbf{\frac{x^4 y^6}{z^8}}$
(v) $(15x^2 y^3)^0 = \mathbf{1}$
(vi) $3^{-3} = \frac{1}{3^3} = \mathbf{\frac{1}{27}}$
(vii) $\frac{x^{-4}}{y^{-5}} = \mathbf{\frac{y^5}{x^4}}$
(viii) $(-1)^{100} + (-1)^{101} = 1 + (-1) = \mathbf{0}$
(ii) $(x^5)^3 = x^{5 \times 3} = \mathbf{x^{15}}$
(iii) $(a^{-2})^{-4} = a^{(-2)(-4)} = \mathbf{a^8}$
(iv) $\left(\frac{x^2 y^3}{z^4}\right)^2 = \mathbf{\frac{x^4 y^6}{z^8}}$
(v) $(15x^2 y^3)^0 = \mathbf{1}$
(vi) $3^{-3} = \frac{1}{3^3} = \mathbf{\frac{1}{27}}$
(vii) $\frac{x^{-4}}{y^{-5}} = \mathbf{\frac{y^5}{x^4}}$
(viii) $(-1)^{100} + (-1)^{101} = 1 + (-1) = \mathbf{0}$
Exercise 6.3 • Polynomial Long Division & Factor Verification
Q1 to Q6. Divide polynomials by monomials and linear binomials:
1. Divide $12x^4 - 8x^3 + 4x^2$ by $4x^2$:
$$= \frac{12x^4}{4x^2} - \frac{8x^3}{4x^2} + \frac{4x^2}{4x^2} = \mathbf{3x^2 - 2x + 1}$$
2. Divide $x^2 + 7x + 12$ by $x + 3$:
$$x^2 + 7x + 12 = (x + 3)(x + 4) \implies \mathbf{\text{Quotient} = x + 4, \text{Remainder} = 0}$$
3. Divide $x^2 - 5x + 6$ by $x - 2$:
$$x^2 - 5x + 6 = (x - 2)(x - 3) \implies \mathbf{\text{Quotient} = x - 3, \text{Remainder} = 0}$$
4. Divide $2x^2 + 5x - 3$ by $2x - 1$:
$$2x^2 + 5x - 3 = (2x - 1)(x + 3) \implies \mathbf{\text{Quotient} = x + 3, \text{Remainder} = 0}$$
5. Divide $x^3 - 8$ by $x - 2$:
$$\text{Long division}: \mathbf{\text{Quotient} = x^2 + 2x + 4, \text{Remainder} = 0}$$
6. Divide $x^3 + 1$ by $x + 1$:
$$\text{Long division}: \mathbf{\text{Quotient} = x^2 - x + 1, \text{Remainder} = 0}$$
$$= \frac{12x^4}{4x^2} - \frac{8x^3}{4x^2} + \frac{4x^2}{4x^2} = \mathbf{3x^2 - 2x + 1}$$
2. Divide $x^2 + 7x + 12$ by $x + 3$:
$$x^2 + 7x + 12 = (x + 3)(x + 4) \implies \mathbf{\text{Quotient} = x + 4, \text{Remainder} = 0}$$
3. Divide $x^2 - 5x + 6$ by $x - 2$:
$$x^2 - 5x + 6 = (x - 2)(x - 3) \implies \mathbf{\text{Quotient} = x - 3, \text{Remainder} = 0}$$
4. Divide $2x^2 + 5x - 3$ by $2x - 1$:
$$2x^2 + 5x - 3 = (2x - 1)(x + 3) \implies \mathbf{\text{Quotient} = x + 3, \text{Remainder} = 0}$$
5. Divide $x^3 - 8$ by $x - 2$:
$$\text{Long division}: \mathbf{\text{Quotient} = x^2 + 2x + 4, \text{Remainder} = 0}$$
6. Divide $x^3 + 1$ by $x + 1$:
$$\text{Long division}: \mathbf{\text{Quotient} = x^2 - x + 1, \text{Remainder} = 0}$$
Q7 to Q12. Division Algorithm & Applications:
7. If divisor is $x - 1$, quotient is $x^2 + x + 1$ and remainder is $0$, find dividend:
$$\text{Dividend} = (x - 1)(x^2 + x + 1) + 0 = \mathbf{x^3 - 1}$$
8. Find quotient and remainder when $3x^3 - 2x^2 + 5$ is divided by $x + 1$:
$$\text{Quotient} = \mathbf{3x^2 - 5x + 5}, \quad \text{Remainder} = 5 - 5 = \mathbf{0} \quad (\text{Exact divisor})$$
9. What should be added to $x^3 - 3x^2 + 4x - 5$ so that it is exactly divisible by $x - 2$?
$$P(2) = (2)^3 - 3(2)^2 + 4(2) - 5 = 8 - 12 + 8 - 5 = -1 \implies \mathbf{\text{Add } +1}$$
10. Find the value of $k$ if $x - 3$ is a factor of $x^2 - kx + 12$:
$$P(3) = (3)^2 - 3k + 12 = 0 \implies 21 - 3k = 0 \implies \mathbf{k = 7}$$
$$\text{Dividend} = (x - 1)(x^2 + x + 1) + 0 = \mathbf{x^3 - 1}$$
8. Find quotient and remainder when $3x^3 - 2x^2 + 5$ is divided by $x + 1$:
$$\text{Quotient} = \mathbf{3x^2 - 5x + 5}, \quad \text{Remainder} = 5 - 5 = \mathbf{0} \quad (\text{Exact divisor})$$
9. What should be added to $x^3 - 3x^2 + 4x - 5$ so that it is exactly divisible by $x - 2$?
$$P(2) = (2)^3 - 3(2)^2 + 4(2) - 5 = 8 - 12 + 8 - 5 = -1 \implies \mathbf{\text{Add } +1}$$
10. Find the value of $k$ if $x - 3$ is a factor of $x^2 - kx + 12$:
$$P(3) = (3)^2 - 3k + 12 = 0 \implies 21 - 3k = 0 \implies \mathbf{k = 7}$$
Exercise 6.4 • Square of Binomials & Perfect Square Factorization
Q1 & Q2. Expand using $(a \pm b)^2 = a^2 \pm 2ab + b^2$:
(i) $(2x + 3)^2$: $(2x)^2 + 2(2x)(3) + 3^2 = \mathbf{4x^2 + 12x + 9}$
(ii) $(3a - 4b)^2$: $(3a)^2 - 2(3a)(4b) + (4b)^2 = \mathbf{9a^2 - 24ab + 16b^2}$
(iii) $(x^2 + 5)^2$: $(x^2)^2 + 2(x^2)(5) + 5^2 = \mathbf{x^4 + 10x^2 + 25}$
(iv) $\left(x - \frac{1}{x}\right)^2$: $x^2 - 2(x)\left(\frac{1}{x}\right) + \frac{1}{x^2} = \mathbf{x^2 - 2 + \frac{1}{x^2}}$
(ii) $(3a - 4b)^2$: $(3a)^2 - 2(3a)(4b) + (4b)^2 = \mathbf{9a^2 - 24ab + 16b^2}$
(iii) $(x^2 + 5)^2$: $(x^2)^2 + 2(x^2)(5) + 5^2 = \mathbf{x^4 + 10x^2 + 25}$
(iv) $\left(x - \frac{1}{x}\right)^2$: $x^2 - 2(x)\left(\frac{1}{x}\right) + \frac{1}{x^2} = \mathbf{x^2 - 2 + \frac{1}{x^2}}$
Q3 & Q4. Numerical Evaluation & Factorization:
(i) Evaluate $(103)^2$: $(100 + 3)^2 = 10000 + 2(100)(3) + 9 = 10000 + 600 + 9 = \mathbf{10,609}$
(ii) Evaluate $(196)^2$: $(200 - 4)^2 = 40000 - 2(200)(4) + 16 = 40000 - 1600 + 16 = \mathbf{38,416}$
(iii) Factorize $4x^2 + 12x + 9$: $(2x)^2 + 2(2x)(3) + 3^2 = \mathbf{(2x + 3)^2}$
(iv) Factorize $25a^2 - 70ab + 49b^2$: $(5a)^2 - 2(5a)(7b) + (7b)^2 = \mathbf{(5a - 7b)^2}$
(ii) Evaluate $(196)^2$: $(200 - 4)^2 = 40000 - 2(200)(4) + 16 = 40000 - 1600 + 16 = \mathbf{38,416}$
(iii) Factorize $4x^2 + 12x + 9$: $(2x)^2 + 2(2x)(3) + 3^2 = \mathbf{(2x + 3)^2}$
(iv) Factorize $25a^2 - 70ab + 49b^2$: $(5a)^2 - 2(5a)(7b) + (7b)^2 = \mathbf{(5a - 7b)^2}$
Exercise 6.5 • Difference of Two Squares $(a^2 - b^2)$
Factorize and Evaluate using $(a+b)(a-b) = a^2 - b^2$:
1. $x^2 - 49$: $(x)^2 - (7)^2 = \mathbf{(x + 7)(x - 7)}$
2. $4a^2 - 25b^2$: $(2a)^2 - (5b)^2 = \mathbf{(2a + 5b)(2a - 5b)}$
3. $16x^4 - 81$: $(4x^2)^2 - 9^2 = (4x^2 + 9)(4x^2 - 9) = \mathbf{(4x^2 + 9)(2x + 3)(2x - 3)}$
4. $3x^2 - 75$: $3(x^2 - 25) = \mathbf{3(x + 5)(x - 5)}$
5. Evaluate $101 \times 99$: $(100 + 1)(100 - 1) = 100^2 - 1^2 = 10000 - 1 = \mathbf{9,999}$
6. Evaluate $502 \times 498$: $(500 + 2)(500 - 2) = 500^2 - 2^2 = 250000 - 4 = \mathbf{249,996}$
2. $4a^2 - 25b^2$: $(2a)^2 - (5b)^2 = \mathbf{(2a + 5b)(2a - 5b)}$
3. $16x^4 - 81$: $(4x^2)^2 - 9^2 = (4x^2 + 9)(4x^2 - 9) = \mathbf{(4x^2 + 9)(2x + 3)(2x - 3)}$
4. $3x^2 - 75$: $3(x^2 - 25) = \mathbf{3(x + 5)(x - 5)}$
5. Evaluate $101 \times 99$: $(100 + 1)(100 - 1) = 100^2 - 1^2 = 10000 - 1 = \mathbf{9,999}$
6. Evaluate $502 \times 498$: $(500 + 2)(500 - 2) = 500^2 - 2^2 = 250000 - 4 = \mathbf{249,996}$
Exercise 6.6 • Factorization of Four-Term Forms $(a \pm b)^2 - c^2$
Factorize by Grouping into Perfect Square Minus Square:
1. $4x^2 + 4xy + y^2 - 9z^2$: $(2x + y)^2 - (3z)^2 = \mathbf{(2x + y + 3z)(2x + y - 3z)}$
2. $x^2 - 6ax + 9a^2 - 16b^2$: $(x - 3a)^2 - (4b)^2 = \mathbf{(x - 3a + 4b)(x - 3a - 4b)}$
3. $9x^2 - 12x + 4 - 169y^2$: $(3x - 2)^2 - (13y)^2 = \mathbf{(3x - 2 + 13y)(3x - 2 - 13y)}$
4. $1 - x^2 - 2xy - y^2$: $1 - (x^2 + 2xy + y^2) = 1^2 - (x + y)^2 = \mathbf{(1 + x + y)(1 - x - y)}$
5. $y^2 - c^2 + 2cx - x^2$: $y^2 - (x^2 - 2cx + c^2) = y^2 - (x - c)^2 = \mathbf{(y + x - c)(y - x + c)}$
6. $x^4 + 8x^2 + 16 - 25z^2$: $(x^2 + 4)^2 - (5z)^2 = \mathbf{(x^2 + 4 + 5z)(x^2 + 4 - 5z)}$
7. $m^2 - 6mc + 9c^2 - k^2$: $(m - 3c)^2 - k^2 = \mathbf{(m - 3c + k)(m - 3c - k)}$
8. $a^4 - 28a^2 + 36$: $(a^4 - 12a^2 + 36) - 16a^2 = (a^2 - 6)^2 - (4a)^2 = \mathbf{(a^2 + 4a - 6)(a^2 - 4a - 6)}$
9. Find polynomial for $(x + y - 2c)(x + y + 2c)$: $(x + y)^2 - (2c)^2 = \mathbf{x^2 + 2xy + y^2 - 4c^2}$
10. Write $x^2 + 4y^2 - z^2 + 4xy$ as difference of two squares: $(x^2 + 4xy + 4y^2) - z^2 = \mathbf{(x + 2y)^2 - z^2}$
2. $x^2 - 6ax + 9a^2 - 16b^2$: $(x - 3a)^2 - (4b)^2 = \mathbf{(x - 3a + 4b)(x - 3a - 4b)}$
3. $9x^2 - 12x + 4 - 169y^2$: $(3x - 2)^2 - (13y)^2 = \mathbf{(3x - 2 + 13y)(3x - 2 - 13y)}$
4. $1 - x^2 - 2xy - y^2$: $1 - (x^2 + 2xy + y^2) = 1^2 - (x + y)^2 = \mathbf{(1 + x + y)(1 - x - y)}$
5. $y^2 - c^2 + 2cx - x^2$: $y^2 - (x^2 - 2cx + c^2) = y^2 - (x - c)^2 = \mathbf{(y + x - c)(y - x + c)}$
6. $x^4 + 8x^2 + 16 - 25z^2$: $(x^2 + 4)^2 - (5z)^2 = \mathbf{(x^2 + 4 + 5z)(x^2 + 4 - 5z)}$
7. $m^2 - 6mc + 9c^2 - k^2$: $(m - 3c)^2 - k^2 = \mathbf{(m - 3c + k)(m - 3c - k)}$
8. $a^4 - 28a^2 + 36$: $(a^4 - 12a^2 + 36) - 16a^2 = (a^2 - 6)^2 - (4a)^2 = \mathbf{(a^2 + 4a - 6)(a^2 - 4a - 6)}$
9. Find polynomial for $(x + y - 2c)(x + y + 2c)$: $(x + y)^2 - (2c)^2 = \mathbf{x^2 + 2xy + y^2 - 4c^2}$
10. Write $x^2 + 4y^2 - z^2 + 4xy$ as difference of two squares: $(x^2 + 4xy + 4y^2) - z^2 = \mathbf{(x + 2y)^2 - z^2}$
Exercise 6.7 • Cube of Binomials $(a \pm b)^3$
Expand using $(a \pm b)^3 = a^3 \pm 3a^2b + 3ab^2 \pm b^3$:
1. $(3x + 2)^3$: $(3x)^3 + 3(3x)^2(2) + 3(3x)(2)^2 + 2^3 = \mathbf{27x^3 + 54x^2 + 36x + 8}$
2. $(x^2 - 9)^3$: $(x^2)^3 - 3(x^2)^2(9) + 3(x^2)(9)^2 - 9^3 = \mathbf{x^6 - 27x^4 + 243x^2 - 729}$
3. $(4x + 3y)^3$: $(4x)^3 + 3(4x)^2(3y) + 3(4x)(3y)^2 + (3y)^3 = \mathbf{64x^3 + 144x^2y + 108xy^2 + 27y^3}$
4. $(a - 7b)^3$: $a^3 - 3a^2(7b) + 3a(7b)^2 - (7b)^3 = \mathbf{a^3 - 21a^2b + 147ab^2 - 343b^3}$
5. $(y - 2x)^3$: $y^3 - 3y^2(2x) + 3y(2x)^2 - (2x)^3 = \mathbf{y^3 - 6x y^2 + 12x^2 y - 8x^3}$
6. $(2d - 4c^2)^3$: $(2d)^3 - 3(2d)^2(4c^2) + 3(2d)(4c^2)^2 - (4c^2)^3 = \mathbf{8d^3 - 48d^2c^2 + 96dc^4 - 64c^6}$
7. $(5x^2 - 1)^3$: $(5x^2)^3 - 3(5x^2)^2(1) + 3(5x^2)(1)^2 - 1^3 = \mathbf{125x^6 - 75x^4 + 15x^2 - 1}$
8. $(x^2 - 4b^2)^3$: $(x^2)^3 - 3(x^2)^2(4b^2) + 3(x^2)(4b^2)^2 - (4b^2)^3 = \mathbf{x^6 - 12x^4b^2 + 48x^2b^4 - 64b^6}$
9. $(y^2 - c^2)^3$: $(y^2)^3 - 3(y^2)^2(c^2) + 3(y^2)(c^2)^2 - (c^2)^3 = \mathbf{y^6 - 3y^4c^2 + 3y^2c^4 - c^6}$
10. $(p^2q^2 - 3a)^3$: $(p^2q^2)^3 - 3(p^2q^2)^2(3a) + 3(p^2q^2)(3a)^2 - (3a)^3 = \mathbf{p^6q^6 - 9ap^4q^4 + 27a^2p^2q^2 - 27a^3}$
2. $(x^2 - 9)^3$: $(x^2)^3 - 3(x^2)^2(9) + 3(x^2)(9)^2 - 9^3 = \mathbf{x^6 - 27x^4 + 243x^2 - 729}$
3. $(4x + 3y)^3$: $(4x)^3 + 3(4x)^2(3y) + 3(4x)(3y)^2 + (3y)^3 = \mathbf{64x^3 + 144x^2y + 108xy^2 + 27y^3}$
4. $(a - 7b)^3$: $a^3 - 3a^2(7b) + 3a(7b)^2 - (7b)^3 = \mathbf{a^3 - 21a^2b + 147ab^2 - 343b^3}$
5. $(y - 2x)^3$: $y^3 - 3y^2(2x) + 3y(2x)^2 - (2x)^3 = \mathbf{y^3 - 6x y^2 + 12x^2 y - 8x^3}$
6. $(2d - 4c^2)^3$: $(2d)^3 - 3(2d)^2(4c^2) + 3(2d)(4c^2)^2 - (4c^2)^3 = \mathbf{8d^3 - 48d^2c^2 + 96dc^4 - 64c^6}$
7. $(5x^2 - 1)^3$: $(5x^2)^3 - 3(5x^2)^2(1) + 3(5x^2)(1)^2 - 1^3 = \mathbf{125x^6 - 75x^4 + 15x^2 - 1}$
8. $(x^2 - 4b^2)^3$: $(x^2)^3 - 3(x^2)^2(4b^2) + 3(x^2)(4b^2)^2 - (4b^2)^3 = \mathbf{x^6 - 12x^4b^2 + 48x^2b^4 - 64b^6}$
9. $(y^2 - c^2)^3$: $(y^2)^3 - 3(y^2)^2(c^2) + 3(y^2)(c^2)^2 - (c^2)^3 = \mathbf{y^6 - 3y^4c^2 + 3y^2c^4 - c^6}$
10. $(p^2q^2 - 3a)^3$: $(p^2q^2)^3 - 3(p^2q^2)^2(3a) + 3(p^2q^2)(3a)^2 - (3a)^3 = \mathbf{p^6q^6 - 9ap^4q^4 + 27a^2p^2q^2 - 27a^3}$
Exercise 6.8 • Number Patterns & General Terms ($a_n$)
Pattern Deductions & General Term Evaluation:
1. Square Numbers ($a_n = n^2$): $1, 4, 9, 16, 25, 36, 49, 64, 81, 100$
2. Cubic Numbers ($a_n = n^3$): $1, 8, 27, 64, 125, 216, 343, 512, 729, 1000$
3. Triangular Numbers ($a_n = \frac{n(n+1)}{2}$): $1, 3, 6, 10, 15, 21, 28, 36, 45, 55$
4. If $a_n = 4n + 3$, find first 4 terms: $a_1 = 7, a_2 = 11, a_3 = 15, a_4 = 19 \implies \mathbf{7, 11, 15, 19}$
5. If $a_n = n^2 + 1$, find 10th term: $a_{10} = 10^2 + 1 = \mathbf{101}$
6. If $a_n = 2n^3 - 1$, find 5th term: $a_5 = 2(5)^3 - 1 = 2(125) - 1 = \mathbf{249}$
2. Cubic Numbers ($a_n = n^3$): $1, 8, 27, 64, 125, 216, 343, 512, 729, 1000$
3. Triangular Numbers ($a_n = \frac{n(n+1)}{2}$): $1, 3, 6, 10, 15, 21, 28, 36, 45, 55$
4. If $a_n = 4n + 3$, find first 4 terms: $a_1 = 7, a_2 = 11, a_3 = 15, a_4 = 19 \implies \mathbf{7, 11, 15, 19}$
5. If $a_n = n^2 + 1$, find 10th term: $a_{10} = 10^2 + 1 = \mathbf{101}$
6. If $a_n = 2n^3 - 1$, find 5th term: $a_5 = 2(5)^3 - 1 = 2(125) - 1 = \mathbf{249}$
Exercise 6.9 • Arithmetic & Geometric Sequences & Real-World Word Problems
Q1, Q2 & Q3. Sequence Classification & Indicated Terms:
1(i) $a_n = 6n + 5$: $a_1 = 11, a_2 = 17, a_3 = 23 \implies \mathbf{\text{Arithmetic (AP)}, d = 6}$
1(vi) $a_n = 2^n$: $a_1 = 2, a_2 = 4, a_3 = 8 \implies \mathbf{\text{Geometric (GP)}, r = 2}$
2(i) $6, 14, 22, \dots$: $d = 14 - 6 = 8 \implies \mathbf{\text{Arithmetic Sequence}, d = 8}$
2(ii) $4, 12, 36, \dots$: $r = \frac{12}{4} = 3 \implies \mathbf{\text{Geometric Sequence}, r = 3}$
3(i) $a_1 = 10, d = 5$, find $a_{13}$: $a_{13} = 10 + (13 - 1)5 = 10 + 60 = \mathbf{70}$
3(ii) $a_1 = 47, d = -5$, find $a_7$: $a_7 = 47 + (7 - 1)(-5) = 47 - 30 = \mathbf{17}$
3(iv) $a_1 = 12, d = -7$, find $a_{12}$: $a_{12} = 12 + (12 - 1)(-7) = 12 - 77 = \mathbf{-65}$
1(vi) $a_n = 2^n$: $a_1 = 2, a_2 = 4, a_3 = 8 \implies \mathbf{\text{Geometric (GP)}, r = 2}$
2(i) $6, 14, 22, \dots$: $d = 14 - 6 = 8 \implies \mathbf{\text{Arithmetic Sequence}, d = 8}$
2(ii) $4, 12, 36, \dots$: $r = \frac{12}{4} = 3 \implies \mathbf{\text{Geometric Sequence}, r = 3}$
3(i) $a_1 = 10, d = 5$, find $a_{13}$: $a_{13} = 10 + (13 - 1)5 = 10 + 60 = \mathbf{70}$
3(ii) $a_1 = 47, d = -5$, find $a_7$: $a_7 = 47 + (7 - 1)(-5) = 47 - 30 = \mathbf{17}$
3(iv) $a_1 = 12, d = -7$, find $a_{12}$: $a_{12} = 12 + (12 - 1)(-7) = 12 - 77 = \mathbf{-65}$
Q4 to Q10. Real-Life Applied Word Problems:
4. Starting salary Rs. 50,000, annual raise Rs. 2,500 ($a_1 = 50000, d = 2500$):
- (i) 5th year: $a_5 = 50000 + (4)(2500) = \mathbf{\text{Rs. } 60,000}$
- (ii) Common difference $d = \mathbf{\text{Rs. } 2,500}$
- (iii) 10th year: $a_{10} = 50000 + (9)(2500) = \mathbf{\text{Rs. } 72,500}$
5. Fruit packets in carriages: $75, 85, 95, \dots$ ($a_1 = 75, d = 10$):
7th carriage: $a_7 = 75 + (6)(10) = \mathbf{135\text{ packets}}$
6. Even numbers from 10 to 98 ($a_1 = 10, d = 2, a_n = 98$):
$$98 = 10 + (n - 1)2 \implies 88 = 2(n - 1) \implies n - 1 = 44 \implies \mathbf{n = 45\text{ numbers}}$$
7. Car distance per minute: $100\text{ m}, 160\text{ m}, 220\text{ m}, \dots$ ($a_1 = 100, d = 60$):
10th minute: $a_{10} = 100 + (9)(60) = 100 + 540 = \mathbf{640\text{ meters}}$
8. Writer words per day: $890, 760, 630, \dots$ ($a_1 = 890, d = -130$):
- Common difference $d = \mathbf{-130\text{ words}}$
- 6th day: $a_6 = 890 + (5)(-130) = 890 - 650 = \mathbf{240\text{ words}}$
9. Interior angle sum: Triangle $180^\circ$, Quad $360^\circ$, Pent $540^\circ$ ($S_n = (n - 2) \times 180^\circ$):
For Dodecagon ($n = 12$ sides): $S_{12} = (12 - 2) \times 180^\circ = 10 \times 180^\circ = \mathbf{1800^\circ}$
10. Building population by floor: Floor 1 = 48, Floor 2 = 39, Floor 3 = 30 ($a_1 = 48, d = -9$):
- (i) 5th floor: $a_5 = 48 + (4)(-9) = 48 - 36 = \mathbf{12\text{ people}}$
- (ii) Total in building (Floor 6 = 3 people, Floor 7 = 0): Total $= 48 + 39 + 30 + 21 + 12 + 3 = \mathbf{153\text{ people}}$
- (i) 5th year: $a_5 = 50000 + (4)(2500) = \mathbf{\text{Rs. } 60,000}$
- (ii) Common difference $d = \mathbf{\text{Rs. } 2,500}$
- (iii) 10th year: $a_{10} = 50000 + (9)(2500) = \mathbf{\text{Rs. } 72,500}$
5. Fruit packets in carriages: $75, 85, 95, \dots$ ($a_1 = 75, d = 10$):
7th carriage: $a_7 = 75 + (6)(10) = \mathbf{135\text{ packets}}$
6. Even numbers from 10 to 98 ($a_1 = 10, d = 2, a_n = 98$):
$$98 = 10 + (n - 1)2 \implies 88 = 2(n - 1) \implies n - 1 = 44 \implies \mathbf{n = 45\text{ numbers}}$$
7. Car distance per minute: $100\text{ m}, 160\text{ m}, 220\text{ m}, \dots$ ($a_1 = 100, d = 60$):
10th minute: $a_{10} = 100 + (9)(60) = 100 + 540 = \mathbf{640\text{ meters}}$
8. Writer words per day: $890, 760, 630, \dots$ ($a_1 = 890, d = -130$):
- Common difference $d = \mathbf{-130\text{ words}}$
- 6th day: $a_6 = 890 + (5)(-130) = 890 - 650 = \mathbf{240\text{ words}}$
9. Interior angle sum: Triangle $180^\circ$, Quad $360^\circ$, Pent $540^\circ$ ($S_n = (n - 2) \times 180^\circ$):
For Dodecagon ($n = 12$ sides): $S_{12} = (12 - 2) \times 180^\circ = 10 \times 180^\circ = \mathbf{1800^\circ}$
10. Building population by floor: Floor 1 = 48, Floor 2 = 39, Floor 3 = 30 ($a_1 = 48, d = -9$):
- (i) 5th floor: $a_5 = 48 + (4)(-9) = 48 - 36 = \mathbf{12\text{ people}}$
- (ii) Total in building (Floor 6 = 3 people, Floor 7 = 0): Total $= 48 + 39 + 30 + 21 + 12 + 3 = \mathbf{153\text{ people}}$
Review Exercise 6 • Comprehensive Assessment & Brain Buster
Q1. Encircle the correct answer:
(i) $2^5 \times 2^{-3} =$ → (c) $2^2$
(ii) $\left(-\frac{3}{4}\right)^6 \times \left(-\frac{3}{4}\right)^9 =$ → (d) $\left(-\frac{3}{4}\right)^{15}$
(iii) $(\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b}) =$ → (d) $a - b$
(iv) The square of $y - 2x$ is: → (a) $y^2 + 4x^2 - 4xy$
(v) The simplified form of $(a + b) - (a - b)$ is: → (c) $2b$
(vi) Factors of $6^2 - x^2$ are: → (d) $(6 + x)(6 - x)$
(vii) $\left(\frac{2}{5} \times \frac{3}{7}\right)^5 =$ → (b) $\left(\frac{2}{5}\right)^5 \times \left(\frac{3}{7}\right)^5$
(viii) If $3x^3 - 2x^2 + 5$ is divided by $x + 1$, then $x + 1$ is its: → (a) divisor as well as factor
(ix) Next two terms in sequence $2, 4, 6, \dots$ are: → (c) $8, 10$
(x) Next two terms in sequence $1, 8, 27, \dots$ ($n^3$) are: → (c) $64, 125$
(ii) $\left(-\frac{3}{4}\right)^6 \times \left(-\frac{3}{4}\right)^9 =$ → (d) $\left(-\frac{3}{4}\right)^{15}$
(iii) $(\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b}) =$ → (d) $a - b$
(iv) The square of $y - 2x$ is: → (a) $y^2 + 4x^2 - 4xy$
(v) The simplified form of $(a + b) - (a - b)$ is: → (c) $2b$
(vi) Factors of $6^2 - x^2$ are: → (d) $(6 + x)(6 - x)$
(vii) $\left(\frac{2}{5} \times \frac{3}{7}\right)^5 =$ → (b) $\left(\frac{2}{5}\right)^5 \times \left(\frac{3}{7}\right)^5$
(viii) If $3x^3 - 2x^2 + 5$ is divided by $x + 1$, then $x + 1$ is its: → (a) divisor as well as factor
(ix) Next two terms in sequence $2, 4, 6, \dots$ are: → (c) $8, 10$
(x) Next two terms in sequence $1, 8, 27, \dots$ ($n^3$) are: → (c) $64, 125$
Q2. Factorize the following:
- (i) $3x^3 - 18x^2 + 6x$: $3x(x^2 - 6x + 2) = \mathbf{3x(x^2 - 6x + 2)}$
- (ii) $2u^3 - 5u^2 - 6u + 15$: $u^2(2u - 5) - 3(2u - 5) = \mathbf{(2u - 5)(u^2 - 3)}$
- (iii) $(a^2 - 5)^2 - 12(a^2 - 5) + 36$: Let $y = a^2 - 5 \implies y^2 - 12y + 36 = (y - 6)^2 = \mathbf{(a^2 - 11)^2}$
- (iv) $x^4 - 8x^2 + 16$: $(x^2 - 4)^2 = \mathbf{[(x + 2)(x - 2)]^2 = (x + 2)^2 (x - 2)^2}$
- (v) $27x^2 - 198xy^4 + 363y^8$: $3(9x^2 - 66xy^4 + 121y^8) = \mathbf{3(3x - 11y^4)^2}$
- (vi) $(x^2 - 9)^2 + 16(x^2 - 9) + 64$: $(x^2 - 9 + 8)^2 = \mathbf{(x^2 - 1)^2 = (x + 1)^2 (x - 1)^2}$
- (vii) $(a - b)^2 - 4c^2$: $(a - b)^2 - (2c)^2 = \mathbf{(a - b + 2c)(a - b - 2c)}$
- (viii) $a^4 - 64b^4$: $(a^2 + 8b^2)(a^2 - 8b^2) = \mathbf{(a^2 + 8b^2)(a + 2\sqrt{2}b)(a - 2\sqrt{2}b)}$
- (ix) $25\left(x + \frac{5}{4}\right)^2 - 16\left(x + \frac{7}{4}\right)^2$: $\left[5\left(x + \frac{5}{4}\right) + 4\left(x + \frac{7}{4}\right)\right]\left[5\left(x + \frac{5}{4}\right) - 4\left(x + \frac{7}{4}\right)\right] = \left(9x + \frac{53}{4}\right)\left(x - \frac{3}{4}\right)$
- (ii) $2u^3 - 5u^2 - 6u + 15$: $u^2(2u - 5) - 3(2u - 5) = \mathbf{(2u - 5)(u^2 - 3)}$
- (iii) $(a^2 - 5)^2 - 12(a^2 - 5) + 36$: Let $y = a^2 - 5 \implies y^2 - 12y + 36 = (y - 6)^2 = \mathbf{(a^2 - 11)^2}$
- (iv) $x^4 - 8x^2 + 16$: $(x^2 - 4)^2 = \mathbf{[(x + 2)(x - 2)]^2 = (x + 2)^2 (x - 2)^2}$
- (v) $27x^2 - 198xy^4 + 363y^8$: $3(9x^2 - 66xy^4 + 121y^8) = \mathbf{3(3x - 11y^4)^2}$
- (vi) $(x^2 - 9)^2 + 16(x^2 - 9) + 64$: $(x^2 - 9 + 8)^2 = \mathbf{(x^2 - 1)^2 = (x + 1)^2 (x - 1)^2}$
- (vii) $(a - b)^2 - 4c^2$: $(a - b)^2 - (2c)^2 = \mathbf{(a - b + 2c)(a - b - 2c)}$
- (viii) $a^4 - 64b^4$: $(a^2 + 8b^2)(a^2 - 8b^2) = \mathbf{(a^2 + 8b^2)(a + 2\sqrt{2}b)(a - 2\sqrt{2}b)}$
- (ix) $25\left(x + \frac{5}{4}\right)^2 - 16\left(x + \frac{7}{4}\right)^2$: $\left[5\left(x + \frac{5}{4}\right) + 4\left(x + \frac{7}{4}\right)\right]\left[5\left(x + \frac{5}{4}\right) - 4\left(x + \frac{7}{4}\right)\right] = \left(9x + \frac{53}{4}\right)\left(x - \frac{3}{4}\right)$
Q3 to Q6 & Brain Buster:
3. Simplify $(2x + 1)(2x - 1) - (2x - 1)^2$:
$$= (4x^2 - 1) - (4x^2 - 4x + 1) = 4x^2 - 1 - 4x^2 + 4x - 1 = \mathbf{4x - 2 = 2(2x - 1)}$$
4. Simplify $60(a + b)^2 - 15(a - b)^2$:
$$= 15[4(a^2 + 2ab + b^2) - (a^2 - 2ab + b^2)] = 15(3a^2 + 10ab + 3b^2) = \mathbf{15(3a + b)(a + 3b)}$$
5. Simplify $(y - z)^2 - (y + z)^2$:
$$= (y^2 - 2yz + z^2) - (y^2 + 2yz + z^2) = \mathbf{-4yz}$$
6. Find first three terms:
- (i) $a_n = 9n + 1 \implies a_1 = 10, a_2 = 19, a_3 = 28 \implies \mathbf{10, 19, 28}$
- (ii) $a_n = n^5 \implies a_1 = 1, a_2 = 32, a_3 = 243 \implies \mathbf{1, 32, 243}$
🧠 Brain Buster:
Why cannot the Product Rule of Exponents be applied directly to $2^3 \times 3^2$?
Resolution: The Product Rule ($x^m \times x^n = x^{m+n}$) requires identical bases ($x$). Here the bases ($2$ and $3$) and exponents ($3$ and $2$) are both different. Thus, they must be evaluated individually: $2^3 \times 3^2 = 8 \times 9 = \mathbf{72}$.
$$= (4x^2 - 1) - (4x^2 - 4x + 1) = 4x^2 - 1 - 4x^2 + 4x - 1 = \mathbf{4x - 2 = 2(2x - 1)}$$
4. Simplify $60(a + b)^2 - 15(a - b)^2$:
$$= 15[4(a^2 + 2ab + b^2) - (a^2 - 2ab + b^2)] = 15(3a^2 + 10ab + 3b^2) = \mathbf{15(3a + b)(a + 3b)}$$
5. Simplify $(y - z)^2 - (y + z)^2$:
$$= (y^2 - 2yz + z^2) - (y^2 + 2yz + z^2) = \mathbf{-4yz}$$
6. Find first three terms:
- (i) $a_n = 9n + 1 \implies a_1 = 10, a_2 = 19, a_3 = 28 \implies \mathbf{10, 19, 28}$
- (ii) $a_n = n^5 \implies a_1 = 1, a_2 = 32, a_3 = 243 \implies \mathbf{1, 32, 243}$
🧠 Brain Buster:
Why cannot the Product Rule of Exponents be applied directly to $2^3 \times 3^2$?
Resolution: The Product Rule ($x^m \times x^n = x^{m+n}$) requires identical bases ($x$). Here the bases ($2$ and $3$) and exponents ($3$ and $2$) are both different. Thus, they must be evaluated individually: $2^3 \times 3^2 = 8 \times 9 = \mathbf{72}$.
More Chapter Notes for Class 8 (FBISE)
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Chapter 13: Probability — Comprehensive Solved Notes & Solution Manual
Self-Assessment Practice
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Test Your Knowledge on Chapter 6: Class 8 Mathematics - Ch 6: Algebra & Sequence Mastery Guide (FBISE)
Practice textbook-aligned solved MCQs with instant answer feedback, step-by-step solutions, and timed test simulation.
⏱️ Class 8 Model Examination
Federal Board of Intermediate and Secondary Education (FBISE)
Comprehensive Proctored Simulation
Class 8 Mathematics - Ch 6: Algebra and Sequence Chapter Mock Test
Test your complete conceptual mastery across all chapters under real board exam conditions with official timer, anti-cheat surveillance, and instant grading.
🕒 35 Mins
📝 25 Questions
🎯 Passing: 50.0%