NCERT Solutions Class 8 Maths Chapter 8 Algebraic Expressions and Identities

Chapter 8 of Class 8 Maths, "Algebraic Expressions and Identities", is an important topic that forms the foundation for understanding algebraic concepts in higher studies. This chapter introduces students to the principles of algebraic expressions, their operations, and numerous standard identities. Mastering these concepts is essential as they play a significant role in solving complex equations and mathematical problems encountered in advanced studies.

The NCERT Solutions for Class 8 Maths Chapter 8 provides detailed explanations and step-by-step solutions to textbook exercises, guiding students through the process of working with algebraic expressions and applying identities effectively. This blog aims to help students enhance their algebraic skills, boost their problem-solving abilities, and gain confidence in their math knowledge as they progress in their academic journey.

1.0Download Class 8 Maths Chapter 8 NCERT Solutions PDF Online

Downloading the NCERT Solutions of Class 8 Chapter 8 Algebraic Expressions and Identities PDF will help you understand the curriculum's key concepts. Below is the link to download the NCERT solutions for Class 8 Maths Chapter 8 PDF.

2.0Class 8 Maths Chapter 8: Breakdown of Exercises

3.0NCERT Questions with Solutions for Class 8 Maths Chapter 8 - Detailed Solutions

Exercise : 8.1

• Add the following : (i) $ab-bc,bc-ca,ca-ab$ (ii) $a-b+ab,b-c+bc,c-a+ac$ (iii) $2p^2{q}^2-3pq+4,5+7pq-3p^2{q}^2$ (iv) ${\ell }^2+{\mathrm{m}}^2,{\mathrm{m}}^2+{\mathrm{n}}^2,{\mathrm{n}}^2+{\ell }^2,2\ell \mathrm{m}+2\mathrm{mn}+2\mathrm{n}\ell$ Sol. (i) Writing the given expressions in separate rows with like terms one below the other, we have $$\begin{gathered} \mathrm{ab}-\mathrm{bc} \\ +\mathrm{bc}-\mathrm{ca} \\ -\mathrm{ab}+\mathrm{ca} \\ 0+0+0 \end{gathered}$$

(ii) Writing the given expressions in separate rows with like terms one below the other, we have

or $ab+bc+ac$ (iii) Writing the given expressions in separate rows with like terms one below the other, we have

$\begin{array}{r}2{\mathrm{p}}^2{\mathrm{q}}^2-3\mathrm{pq}+4\\ -3{\mathrm{p}}^2{\mathrm{q}}^2+7\mathrm{pq}+5\\ -{\mathrm{p}}^2{\mathrm{q}}^2+4\mathrm{pq}+9\end{array}$

(iv) Writing the given expressions in separate rows with like terms one below the other, we have ${\ell }^2+{\mathrm{m}}^2$ $+{\mathrm{m}}^2+{\mathrm{n}}^2$ ${\ell }^2+{\mathrm{n}}^2$ $\frac{+2\ell m+2mn+2n\ell }{\underline{2{\ell }^2+2m^2+2n^2+2\ell m+2mn+2n\ell }}$

• (i) Subtract $4a-7ab+3b+12$ from $12a$ $-9ab+5b-3$ (ii) Subtract $3xy+5yz-7zx$ from $5xy-$ $2yz-2zx+10xyz$ (iii) Subtract $4p^{2}q-3pq+5p{q}^2-8p+7q$ -10 from $18-3p-11q+5pq-2q^2$ $+5p^{2}q$. Sol. Rearranging the terms of the given expressions, changing the sign of each term of the expression to be subtracted and adding the two expressions, we get, (i) $\begin{array}{r}12\mathrm{a}-9\mathrm{ab}+5\mathrm{b}-3\\ 4\mathrm{a}-7\mathrm{ab}+3\mathrm{b}+12\end{array}$

(ii) Rearranging the terms of the given expressions, changing the sign of each term of the expression to be subtracted and adding the two expressions, we get,

(iii) Rearranging the terms of the given expressions, changing the sign of each term of the expression to be subtracted and adding the two expressions, we get. $5p^{2}q-2p{q}^2+5qp-11q-3p+18$ $4p^{2}q+5q^2-3pq+7q-8p-10$ $\frac{--+-++}{p^{2}q-7q^2+8pq-18q+5p+28}$

• Identify the terms, their coefficients for each of the following expressions : (i) $5x_{yz}{}^2-3zy$ (ii) $1+x+x^2$ (iii) $4x^2{y}^2-4x^2{y}^2{z}^2+z^2$ (iv) $3-\mathrm{pq}+\mathrm{qr}-\mathrm{rp}$ (v) $\frac{x}{2}+\frac{y}{2}-xy$ (vi) $0.3\mathrm{a}-0.6\mathrm{ab}+0.5\mathrm{b}$ Sol. (i) $5xy{z}^2-3zy$ In the expression $5x^2{z}^2-3zy$, the terms are $5x^{\prime 2}{z}^2$ and $-3zy$. Coefficient of $xy{z}^2$ in the term $5x^2{z}^2$ is 5 . Coefficient of zy in the term $-3zy$ is -3 . (ii) $1+x+x^2$ In the expression $1+x+x^2$, the terms are $1,\mathrm{x}$ and ${\mathrm{x}}^2$. Coefficient of $x^0$ in the term 1 is 1 . Coefficient of x in the term x is 1 . Coefficient of $x^2$ is the term $x^2$ is 1 . (iii) In the expression $4x^2{y}^2-4x^2{y}^2{z}^2+z^2$, the terms are $4x^2{y}^2,-4x^2{y}^2{z}^2$ and $z^2$. Coefficient of $x^2{y}^2$ in the term $4x^2{y}^2$ is 4 . Coefficient of $x^2{y}^2{z}^2$ in the term $-4x^2{y}^2{z}^2$ is -4 . Coefficient of $z^2$ in the term $z^2$ is 1 . (iv) 3 - pq + qr - rp In the expression $3-\mathrm{pq}+\mathrm{qr}-\mathrm{rp}$, the terms are 3, -pq, qr and -rp Coefficient of $x^0$ in the term 3 is 3 . Coefficient of pq in the term -pq is -1 . Coefficient of qr in the term qr is 1. Coefficient of rp in the term -rp is -1 . (v) In the expression $\frac{x}{2}+\frac{y}{2}-xy$, the terms are $\frac{x}{2},\frac{y}{2}$ and $-xy$ Coefficient of $x$ in the term $\frac{x}{2}$ is $\frac{1}{2}$. Coefficient of $y$ in the term $\frac{y}{2}$ is $\frac{1}{2}$. Coefficient of xy in the term -xy is -1 . (vi) In the expression $0.3a-0.6ab+0.5b$, the terms are $0.3\mathrm{a},-0.6\mathrm{ab}$ and 0.5 b Coefficient of a in the term 0.3 a is 0.3 . Coefficient of ab in the term -0.6 ab is - 0.6. Coefficient of $b$ in the term $0.5b$ is 0.5 .
• Classify the following polynomials as monomials, binomials, trinomials. Which polynomials do not fit in any of these three categories? $x+y,1000,x+x^2+x^3+x^4,7+y+5x,2y-$ $3y^2,2y-3y^2+4y^3,5x-4y+3xy,4z-15z^2$, $ab+bc+cd+da,pqr,p^{2}q+q^2,2p+2q$. Sol. The given polynomials are classified as under: Monomials : 1000, pqr Binomials : $x+y,2y-3y^2,4z-15z^2,p^{2}q+$ ${\mathrm{pq}}^2,2\mathrm{p}+2\mathrm{q}$. Trinomials: $7+y+5x,2y-3y^2+4y^3,5x$ $-4y+3xy$. Polynomials that do not fit in any category : $x+x^2+x^3+x^4,ab+bc+cd+da$

Exercise : 8.2

• Find the product of the following pairs of monomials (i) $4,7\mathrm{p}$ (ii) $-4p,7p$ (iii) $-4\mathrm{p},7\mathrm{pq}$ (iv) $4p^3,-3p$ (v) $4\mathrm{p},0$ Sol. (i) $4\times 7\mathrm{p}=(4\times 7)\times \mathrm{p}=28\mathrm{p}$ (ii) $-4p\times 7p=(-4\times 7)\times (p\times p)$ $=-28p^{1+1}$ $=-28p^2$ (iii) $-4p\times 7pq=(-4\times 7)\times (p\times p\times q)$ $=-28p^{1+1}q$ $=-28p^{2}q$ (iv) $4p^3\times (-3p)=\{4\times (-3)\}\times \left(p^3\times p\right)$ $=-12\times p^{3+1}$ $=-12p^4$ (v) $4\mathrm{p}\times 0=(4\times 0)\times (\mathrm{p})$ $=0\times \mathrm{p}$ $=0$
• Find the areas of rectangles with the following pairs of monomials as their lengths and breadths respectively : (p, q); (10m, 5n); $\left(20x^2,5y^2\right)$; (4x, $3x^2$ ); (3mn, 4np) Sol. We know that the area of a rectangle $=\ell \times \mathrm{b}$, where $\ell =$ length and $\mathrm{b}=$ breadth . Therefore, the areas of rectangles with pair of monomials (p, q); (10m, 5n); $\left(20x^2,5y^2\right)$; $\left(4x,3x^2\right)$ and ( $3\mathrm{mn},4np$ ) as their lengths and breadths are given by $p\times q=pq$ $10\mathrm{m}\times 5\mathrm{n}=(10\times 5)\times (\mathrm{m}\times \mathrm{n})=50\mathrm{mn}$ $20x^2\times 5y^2=(20\times 5)\times \left(x^2\times y^2\right)$ $=100{\mathrm{x}}^2{\mathrm{y}}^2$ $4\mathrm{x}\times 3{\mathrm{x}}^2=(4\times 3)\times \left(\mathrm{x}\times {\mathrm{x}}^2\right)$ $=12x^3$ and, $3\mathrm{mn}\times 4\mathrm{np}=(3\times 4)\times (\mathrm{m}\times \mathrm{n}\times \mathrm{n}\times \mathrm{p})$ $=12{\mathrm{mn}}^2\mathrm{p}$
• Complete the table of products :

	2x	$-5y$	$3{\mathrm{x}}^2$	$-4xy$	$7x^{2}y$	$-9x^2{y}^2$
2x	$4{\mathrm{x}}^2$	-	-	-	-	-
-5y	-	-	$-15x^{2}y$	-	-	-
$3x^2$	-	-	-	-	-	-
$-4xy$	-	-	-	-	-	-
$7x^{2}y$	-	-	-	-	-	-
$-9x^2{y}^2$	-	-	-	-	-	-

Sol.

• Obtain the volume of rectangular boxes with the following length, breadth and height respectively : (i) $5\mathrm{a},3{\mathrm{a}}^2,7{\mathrm{a}}^4$ (ii) $2\mathrm{p},4\mathrm{q},8\mathrm{r}$ (iii) $xy,2x^{2}y,2x{y}^2$ (iv) a, 2b, 3c Sol. (i) Required volume $=5\mathrm{a}\times 3{\mathrm{a}}^2\times 7{\mathrm{a}}^4$ $=(5\times 3\times 7)\times \left(a\times a^2\times a^4\right)$ $=105a^{1+2+4}=105a^7$ (ii) Required volume $=2p\times 4q\times 8r$ $=(2\times 4\times 8)\times (p\times q\times r)$ $=64\mathrm{pqr}$ (iii) Required volume $=xy\times 2x^{2}y\times 2x{y}^2$ $=(1\times 2\times 2)\times \left(xy\times x^{2}y\times x^2\right)$ $=(4)\times \left(x^{1+2+1}\times y^{1+1+2}\right)$ $=4x^4{y}^4$ (iv) Required volume $=a\times 2b\times 3c$ $=(1\times 2\times 3)\times (a\times b\times c)$ $=6\mathrm{abc}$
• Obtain the product of (i) $xy,yz,zx$ (ii) $\mathrm{a},-{\mathrm{a}}^2,{\mathrm{a}}^3$ (iii) $2,4y,8y^2,16y^3$ (iv) a, 2b, 3c, 6abc (v) $\mathrm{m},-\mathrm{mn},\mathrm{mnp}$ Sol. (i) $\mathrm{xy}\times \mathrm{yz}\times \mathrm{zx}=\mathrm{x}\times \mathrm{x}\times \mathrm{y}\times \mathrm{y}\times \mathrm{z}\times \mathrm{z}$ $=x^{1+1}\times y^{1+1}\times z^{1+1}$ $=x^2{y}^2{z}^2$ (ii) $a\times \left(-a^2\right)\times a^3$ $=[1\times (-1)\times 1]\times (a\times a\times a\times a\times a\times a)$ $=(-1)\times \left(a^6\right)$ $=-{\mathrm{a}}^6$ (iii) $2\times (4y)\times 8y^2\times 16y^3$ $=(2\times 4\times 8\times 16)\times \left(y\times y^2\times y^3\right)$ $=(1024)\times \left(y^{1+2+3}\right)$ $=1024{\mathrm{y}}^6$ (iv) $\mathrm{a}\times 2\mathrm{b}\times 3\mathrm{c}\times 6\mathrm{abc}$ $=(2\times 3\times 6)\times (a\times b\times c\times abc)$ $=(36)\times \left({\mathrm{a}}^{1+1}\times {\mathrm{b}}^{1+1}\times {\mathrm{c}}^{1+1}\right)$ $=36a^2{b}^2{c}^2$ (v) $\mathrm{m}\times -\mathrm{mn}\times \mathrm{mnp}$ $=(1\times -1\times 1)\times (m\times m\times m\times n\times n\times p)$ $=-1\times {\mathrm{m}}^3\times {\mathrm{n}}^2\times \mathrm{p}$ $=-m^3{n}^{2}p$

Exercise : 8.3

• Carry out the multiplication of the expressions in each of the following pairs: (i) $4p,q+r$ (ii) ab, a - b (iii) $a+b,7a^2{b}^2$ (iv) $a^2-9,4a$ (v) $\mathrm{pq}+\mathrm{qr}+\mathrm{rp},0$ Sol. (i) $4p\times (q+r)=4p\times q+4p\times r$ $=4\mathrm{pq}+4\mathrm{pr}$ (ii) $ab\times (a-b)=ab\times a-ab\times (b)$ $=a^{2}b-a{b}^2$ (iii) $(a+b)\times \left(7a^2{b}^2\right)=7a^2{b}^2\times a+7a^2{b}^2\times b$ $=7{\mathrm{a}}^3{\mathrm{b}}^2+7{\mathrm{a}}^2{\mathrm{b}}^3$ (iv) $\left(a^2-9\right)\times 4a=a^2\times 4a-9\times 4a$ $=4a^3-36a$ (v) $(\mathrm{pq}+\mathrm{qr}+\mathrm{rp})\times 0=0$
• Complete the table:

Sol. (i) $a(b+c+d)=a\times b+a\times c+a\times d=ab$ $+ac+ad$ (ii) $(x+y-5)\times (5xy)$ $=5xy\times x+5xy\times y+5xy\times (-5)$ $=5x^{2}y+5x{y}^2-25xy$ (iii) $6p^3-7p^2+5p$ (iv) $4p^4{q}^2-4p^2{q}^4$ (v) $a^{2}bc+a{b}^{2}c+ab{c}^2$

• Find the product: (i) $\left({\mathrm{a}}^2\right)\times \left(2{\mathrm{a}}^{22}\right)\times \left(4{\mathrm{a}}^{26}\right)$ (ii) $\left(\frac{2}{3}xy\right)\times \left(\frac{-9}{10}{x}^2{y}^2\right)$ (iii) $\left(-\frac{10}{3}{\mathrm{pq}}^3\right)\times \left(\frac{6}{5}{\mathrm{p}}^3\mathrm{q}\right)$ (iv) $x\times x^2\times x^3\times x^4$ Sol. (i) $\left({\mathrm{a}}^2\right)\times \left(2{\mathrm{a}}^{22}\right)\times \left(4{\mathrm{a}}^{26}\right)=(1\times 2\times 4)\times$ $\left(a^2\times a^{22}\times a^{26}\right)=8a^2+22+26=8a^{50}$ (ii) $\left(\frac{2}{3}xy\right)\times \left(\frac{-9}{10}{x}^2{y}^2\right)$ $=\left(\frac{2}{3}\times \frac{-9}{10}\right)\times \left(x\times x^2\times y\times y^2\right)$ $=-\frac{3}{5}\times x^{1+2}\times y^{1+2}$ $=-\frac{3}{5}{\mathrm{x}}^3{\mathrm{y}}^3$ (iii) $\left(-\frac{10}{3}{\mathrm{pq}}^3\right)\times \left(\frac{6}{5}{\mathrm{p}}^3\mathrm{q}\right)$ $=\left(-\frac{10}{3}\times \frac{6}{5}\right)\times \left(\mathrm{p}\times {\mathrm{q}}^3\times {\mathrm{p}}^3\times \mathrm{q}\right)$ $=(-4)\times \left(p^4\times q^4\right)=-4p^4{q}^4$ (iv) $\mathrm{x}\times {\mathrm{x}}^2\times {\mathrm{x}}^3\times {\mathrm{x}}^4={\mathrm{x}}^{1+2+3+4}={\mathrm{x}}^{10}$
• (a) Simplify: $3x(4x-5)+3$ and find its values for (i) $\mathrm{x}=3$, (ii) $\mathrm{x}=\frac{1}{2}$. (b) Simplify: $a\left(a^2+a+1\right)+5$ and find its value for (i) $\mathrm{a}=0$, (ii) $\mathrm{a}=1$, (iii) $\mathrm{a}=-1$. Sol. (a) We have, $3\mathrm{x}(4\mathrm{x}-5)+3=3\mathrm{x}\times 4\mathrm{x}-3\mathrm{x}\times 5+3$ $=12x^2-15x+3$ (i) When $x=3$, then $12(3{)}^2-15(3)+3=66$ (ii) When $\mathrm{x}=\frac{1}{2}$, then $$\begin{gathered} 12{\left(\frac{1}{2}\right)}^2-15\times \frac{1}{2}+3 \\ =3-\frac{15}{2}+3=-\frac{3}{2} \end{gathered}$$ (b) We have, $a\left(a^2+a+1\right)+5$ $=a\times a^2+a\times a+a\times 1+5$ $=a^3+a^2+a+5$ (i) When $a=0$, then $a^3+a^2+a+5$ $=0^3+0^2+0+5$ $=5$ (ii) When $a=1$, then $a^3+a^2+a+5$ $1^3+1^2+1+5$ $=1+1+1+5$ $=3+5=8$ (iii) When $\mathrm{a}=-1$, then ${\mathrm{a}}^3+{\mathrm{a}}^2+\mathrm{a}+5$ $=(-1{)}^3+(-1{)}^2+(-1)+5$ $=-1+1-1+5=4$
• (a) Add: $\mathrm{p}(\mathrm{p}-\mathrm{q}),\mathrm{q}(\mathrm{q}-\mathrm{r})$ and $\mathrm{r}(\mathrm{r}-\mathrm{p})$ (b) Add: $2\mathrm{x}(\mathrm{z}-\mathrm{x}-\mathrm{y})$ and $2\mathrm{y}(\mathrm{z}-\mathrm{y}-\mathrm{x})$ (c) Subtract: $3\ell (\ell -4m+5n)$ from $4\ell (10n-3m+2\ell )$ (d) Subtract: $3\mathrm{a}(\mathrm{a}+\mathrm{b}+\mathrm{c})-2\mathrm{b}(\mathrm{a}-\mathrm{b}+\mathrm{c})$ from $4c(-a+b+c)$ Sol. (a) $p(p-q)+q(q-r)+r(r-p)$ $=p\times p-p\times q+q\times q-q\times r+r\times r$ $-\mathrm{r}\times \mathrm{p}$ $=p^2-pq+q^2-qr+r^2-pr$ $=p^2+q^2+r^2-pq-qr-pr$ (b) $2x(z-x-y)+2y(z-y-x)$ $=(2x\times z)-(2x\times x)-(2x\times y)+(2y\times z)$ $-(2y\times y)-(2y\times x)$ $=2\mathrm{xz}-2{\mathrm{x}}^2-2\mathrm{xy}+2\mathrm{yz}-2{\mathrm{y}}^2-2\mathrm{xy}$ $=-2x^2-2y^2-4xy+2xz+2yz$. (c) $4\ell (10n-3m+2\ell )-3\ell (\ell -4m+5n)$ $=(4\ell \times 10\mathrm{n})-(4\ell \times 3\mathrm{m})+(4\ell \times 2\ell )+$ $(-3\ell \times \ell )+(-3\ell \times -4\mathrm{m})+(-3\ell \times 5\mathrm{n})$ $=40\ell \mathrm{n}-12\ell \mathrm{m}+8{\ell }^2-3{\ell }^2+12\ell \mathrm{m}-15\ell \mathrm{n}$ $=5{\ell }^2+25\ell n$ (d) $4\mathrm{c}(-\mathrm{a}+\mathrm{b}+\mathrm{c})-\{3\mathrm{a}(\mathrm{a}+\mathrm{b}+\mathrm{c})-2\mathrm{b}(\mathrm{a}-\mathrm{b}+\mathrm{c})\}$ $=\{4c\times (-a)+4c\times b+4c\times c\}-[\{(3a$ $\times a+3a\times b+3a\times c)\}+\{(-2b\times a)+$ $(-2b\times -b)+(-2b\times c)\}]$ $=-4ac+4bc+4c^2-[3a^2+3ab+3ac$ $-2ab+2b^2-2bc]$ $=-4ac+4bc+4c^2-3a^2-3ab-3ac+$ $2\mathrm{ab}-2{\mathrm{b}}^2+2\mathrm{bc}$ $=-3a^2-2b^2+4c^2-7ac+6bc-ab$

Exercise: 8.4

• 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 $4\left(a^2-\frac{2}{3}{b}^2\right)$ $=\left(\frac{3}{4}{a}^2+3b^2\right)\times \left(4a^2-\frac{8}{3}{b}^2\right)$ $=\frac{3}{4}{\mathrm{a}}^2\left(4{\mathrm{a}}^2-\frac{8}{3}{\mathrm{b}}^2\right)+3{\mathrm{b}}^2\left(4{\mathrm{a}}^2-\frac{8}{3}{\mathrm{b}}^2\right)$ $=3a^4-2a^2{b}^2+12a^2{b}^2-8b^4$ $=3a^4+10a^2{b}^2-8b^4$
• 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) $\left(p^2-q^2\right)(2p+q)=p^2(2p+q)-q^2(2p+q)$ $=2p^3+p^{2}q-2p{q}^2-q^3$
• 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) $(\mathrm{a}+\mathrm{b}+\mathrm{c})(\mathrm{a}+\mathrm{b}-\mathrm{c})$ $=a(a+b-c)+b(a+b-c)+c(a+b-c)$ $=a^2+ab-ac+ab+b^2-bc+ac+bc-c^2$ $=a^2+2ab+b^2-c^2$