NCERT Solutions Class 8 Maths Chapter 8 Algebraic Expressions and Identities Exercise 8.4

NCERT Solutions for Class 8 Maths Exercise 8.4 focuses on solving problems using algebraic identities. These identities help simplify and solve expressions easily without expanding them fully. In this part of the chapter, you will learn to apply standard identities quickly solve expressions and equations.

Our NCERT Solutions for Class 8 Maths Exercise 8.4 are designed to help students understand how to apply these identities in a step-by-step and simple manner. These solutions are perfect for clearing doubts, improving speed, and preparing for exams confidently. With clear explanations and examples, students can strengthen their foundation in algebra.

1.0Download NCERT Solutions Class 8 Maths Chapter 8 Algebraic Expressions and Identities Exercise 8.4: Free PDF

Download the free NCERT Solutions Class 8 Maths Chapter 8 Algebraic Expressions and Identities Exercise 8.4 PDF with easy, step-by-step solutions to help you learn and revise quickly.

2.0Key Concepts in Exercise 8.4 of Class 8 Maths Chapter 8

Exercise 8.4 focuses on the multiplication of algebraic expressions involving binomials, helping students understand how to apply the distributive property and simplify algebraic products.

• Multiplying a Monomial by a Binomial
  Students learn how to multiply a single-term expression (monomial) with a two-term expression (binomial) using the distributive property.
• Using Distributive Law
  The formula used is: $a(b+c)=ab+ac$
  This helps in breaking down expressions and simplifying them step-by-step.
• Multiplying a Binomial by a Binomial
  The exercise also introduces how to multiply two binomials by applying the distributive property twice.
• Combining Like Terms
  After multiplication, students simplify the resulting expressions by combining like terms.
• Building Foundation for Identities
  This exercise builds the groundwork for understanding standard algebraic identities covered in the next section.

3.0NCERT Class 8 Maths Chapter 8: Other Exercises

4.0NCERT Class 8 Maths Chapter 8 Exercise 8.4: Detailed Solutions

1. Multiply the binomials: (i) $(2x+5)$ and $(4x-3)$ (ii) $(y-8)$ and $(3y-4)$ (iii) $(2.5\ell -0.5\mathrm{m})$ and $(2.5\ell +0.5\mathrm{m})$ (iv) $(\mathrm{a}+3\mathrm{b})$ and $(\mathrm{x}+5)$ (v) $\left(2pq+3q^2\right)$ and $\left(3pq-2q^2\right)$ (vi) $\left(\frac{3}{4}{a}^2+3b^2\right)$ and $4\left(a^2-\frac{2}{3}{b}^2\right)$ Sol. (i) $(2x+5)$ and $(4x-3)$ $=2x(4x-3)+5(4x-3)$ $=8{\mathrm{x}}^2-6\mathrm{x}+20\mathrm{x}-15=8{\mathrm{x}}^2+14\mathrm{x}-15$ (ii) $(y-8)\times (3y-4)=y(3y-4)-8(3y-4)$ $=3y^2-4y-24y+32$ $=3y^2-28y+32$ (iii) $(2.5\ell -0.5\mathrm{m})\times (2.5\ell +0.5\mathrm{m})$ $=2.5\ell (2.5\ell +0.5\mathrm{m})-0.5\mathrm{m}(2.5\ell +0.5\mathrm{m})$ $=6.25{\ell }^2+1.25\ell \mathrm{m}-1.25\ell \mathrm{m}-0.25{\mathrm{m}}^2$ $=6.25{\ell }^2-0.25{\mathrm{m}}^2$ (iv) $(a+3b)\times (x+5)$ $=a(x+5)+3b(x+5)$ $=ax+5a+3bx+15b$ $=ax+3bx+5a+15b$ (v) $\left(2pq+3q^2\right)\times \left(3pq-2q^2\right)$ $=2pq\left(3pq-2q^2\right)+3q^2\left(3pq-2q^2\right)$ $=6p^2{q}^2-4p{q}^3+9p{q}^3-6q^4$ $=6p^2{q}^2+5p{q}^3-6q^4$ (vi) $\left(\frac{3}{4}{a}^2+3b^2\right)$ and $4\left(a^2-\frac{2}{3}{b}^2\right)$ $=\left(\frac{3}{4}{a}^2+3b^2\right)\times \left(4a^2-\frac{8}{3}{b}^2\right)$ $=\frac{3}{4}{\mathrm{a}}^2\left(4{\mathrm{a}}^2-\frac{8}{3}{\mathrm{b}}^2\right)+3{\mathrm{b}}^2\left(4{\mathrm{a}}^2-\frac{8}{3}{\mathrm{b}}^2\right)$ $=3a^4-2a^2{b}^2+12a^2{b}^2-8b^4$ $=3a^4+10a^2{b}^2-8b^4$
2. Find the product: (i) $(5-2x)(3+x)$ (ii) $(x+7y)(7x-y)$ (iii) $\left(a^2+b\right)\left(a+b^2\right)$ (iv) $\left(p^2-q^2\right)(2p+q)$ Sol. (i) $(5-2x)(3+x)=5(3+x)-2x(3+x)$ $=15+5x-6x-2x^2$ $=-2x^2-x+15$ (ii) $(x+7y)(7x-y)$ $$\begin{gathered} =x(7x-y)+7y(7x-y) \\ =7x^2-xy+49xy-7y^2 \\ =7x^2+48xy-7y^2 \end{gathered}$$ (iii) $\left(a^2+b\right)\left(a+b^2\right)$ $$\begin{gathered} =a^2\left(a+b^2\right)+b\left(a+b^2\right) \\ =a^3+a^2{b}^2+ab+b^3 \end{gathered}$$ (iv) $\left(p^2-q^2\right)(2p+q)=p^2(2p+q)-q^2(2p+q)$ $=2p^3+p^{2}q-2p{q}^2-q^3$
3. Simplify : (i) $\left(x^2-5\right)(x+5)+25$ (ii) $\left(a^2+5\right)\left(b^3+3\right)+5$ (iii) $\left(t+s^2\right)\left(t^2-s\right)$ (iv) $(\mathrm{a}+\mathrm{b})(\mathrm{c}-\mathrm{d})+(\mathrm{a}-\mathrm{b})(\mathrm{c}+\mathrm{d})+2(\mathrm{ac}+\mathrm{bd})$ (v) $(x+y)(2x+y)+(x+2y)(x-y)$ (vi) $(x+y)\left(x^2-xy+y^2\right)$ (vii) $(1.5x-4y)(1.5x+4y+3)-4.5x+12y$ (viii) $(a+b+c)(a+b-c)$ Sol. (i) $\left(x^2-5\right)(x+5)+25$ $$\begin{gathered} =x^2(x+5)-5(x+5)+25 \\ =x^3+5x^2-5x-25+25 \\ =x^3+5x^2-5x \end{gathered}$$ (ii) $\left(a^2+5\right)\left(b^3+3\right)+5$ $=a^2\left(b^3+3\right)+5\left(b^3+3\right)+5$ $=a^2{b}^3+3a^2+5b^3+15+5$ $=a^2{b}^3+3a^2+5b^3+20$ (iii) $\left(t+s^2\right)\left(t^2-s\right)$ $=t\left(t^2-s\right)+s^2\left(t^2-s\right)$ $={\mathrm{t}}^3-\mathrm{ts}+{\mathrm{s}}^2{\mathrm{t}}^2-{\mathrm{s}}^3$ (iv) $(a+b)(c-d)+(a-b)(c+d)+2(ac+bd)$ $=a(c-d)+b(c-d)+a(c+d)-b(c+d)$ $+2\mathrm{ac}+2\mathrm{bd}$ $=ac-ad+bc-bd+ac+ad-bc-bd$ $+2\mathrm{ac}+2\mathrm{bd}$ $=4\mathrm{ac}$ (v) $(x+y)(2x+y)+(x+2y)(x-y)$ $=x(2x+y)+y(2x+y)+x(x-y)+2y(x-y)$ $=2x^2+xy+2xy+y^2+x^2-xy+2xy-2y^2$ $=(2+1)x^2+(1+2-1+2)xy+(1-2)y^2$ $=3x^2+4xy-y^2$ (vi) $(x+y)\left(x^2-xy+y^2\right)$ $=x\left(x^2-xy+y^2\right)+y\left(x^2-xy+y^2\right)$ $=x^3-x^{2}y+x{y}^2+x^{2}y-x{y}^2+y^3$ $=x^3+y^3$ (vii) $(1.5x-4y)(1.5x+4y+3)-4.5x+12y$ $=1.5x(1.5x+4y+3)-4y(1.5x+4y+3)$ $-4.5x+12y$ $=1.5\mathrm{x}\times 1.5\mathrm{x}+1.5\mathrm{x}\times 4\mathrm{y}+1.5\mathrm{x}\times 3$ $-4y\times 1.5x-4y\times 4y-4y\times 3-4.5x$ $+12\mathrm{y}$ $=2.25x^2+6xy+4.5x-6xy-16y^2$ $-12y-4.5x+12y$ $=2.25x^2+(6-6)xy+(4.5-4.5)x$ $-16y^2+(-12+12)y$ $=2.25x^2+(0)xy+(0)x-16y^2+(0)y$ $=2.25{\mathrm{x}}^2+0+0-16{\mathrm{y}}^2+0$ $=2.25{\mathrm{x}}^2-16{\mathrm{y}}^2$ (viii) $(\mathrm{a}+\mathrm{b}+\mathrm{c})(\mathrm{a}+\mathrm{b}-\mathrm{c})$ $=a(a+b-c)+b(a+b-c)+c(a+b-c)$ $=a^2+ab-ac+ab+b^2-bc+ac+bc-c^2$ $=a^2+2ab+b^2-c^2$

5.0Key Features and Benefits of Class 8 Maths Chapter 8 Exercise 8.4

• Simplifies Complex Concepts: Students learn to expand and simplify these expressions with identities, providing opportunities for faster calculations.
• Strengthens Algebra Skills: Repeated practice improves algebraic skills and promotes understanding and retention of algebra, which is required for advanced algebra topics.
• Step by Step Learning: The questions are put together to promote progressive learning from basic to slightly complex.
• Based on NCERT Guidelines: The exercise follows the NCERT textbook exactly, making it perfect for exam preparation and concept clarity.