ML AGGARWAL CLASS 8 CHAPTER 10 Algebraic Expressions and Identities Exercise 10.3

 Exercise 10.3

Question 1

Sol :

(i) (5 x-2)(3 x+4)

5 x(3 x+4)-2(3 x+4)

$15 x^{2}+20 x-6 x-8$

$15 x^{2}+14 x-8$

(ii) (a x+b)(c x+d)

ax(c x+d)+b(c x+d)

$a c x^{2}+a d x+b c x+b d$

$a c x^{2}+(a t+b c) x+b d$

(iii) (4 p-7)(2-3 p)

4P(2-3 p)=7(2-3 p)

$8 p-12 p^{2}-14+21 P$

$-12 p^{2}+29 p-14$

(iv) $\left(2 x^{2}+3\right)(3 x-5)$

$2 x^{2}(3 x-5)+3(3 x-5)$

$6 x^{3}-10 x^{2}+9 x-15$

$6 x^{3}-10 x^{2}+9 x-15$

(v)  $(1.5 a-2.5 b)(1.5 a+2.5 b)$

$1.5 a(1.5 a+2.5 b)-2.5 b(1.5 a+2.5 b)$

$(1.5 \times 1.5) a^{2}+(1.5 \times 2 \cdot 5) a b-(2.5 \times 1.5) a b-(2.5 \times 2.5) b^{2}$

$2.25 a^{2}+3.75 a b-3.75 a b-6.25 b^{2}$

$2.25 a^{2}+0-6.25 b^{2}$

$2 \cdot 25 a^{2}-6.25 b^{2}$

(vi) $\left(\frac{3}{7} p^{2}+4 q^{2}\right)\left(7\left(p^{2}-\frac{3}{4} q^{2}\right)\right.$

$\frac{3}{7} p^{v} \times\left(7 p^{2}-\frac{21}{4} q^{2}\right)+4 q^{2}\left(-1 p^{2}-\frac{21}{4} q^{2}\right)$

$3 p^{4}-\frac{9}{4} q^{2} p^{2}+28 p^{2} q^{2}-21 q^{4}$

$3 p^{4}+\frac{103}{4} p^{2} q^{2}-21 q^{4}$

Question 2

Sol :
(i) (x-2 y+3)(x+2 y)

$x^{2}+3 x+6 y-4 y^{2}$

(ii) $\left(3-5 x+2 x^{2}\right)(4 x-5)$

Question 3

Sol :
(i) $\left(3 x^{2}-2 x-1\right)\left(2 x^{2}+x-5\right)$

(ii) $\left(2-3 y-5 y^{2}\right)\left(2 y-1+3 y^{2}\right)$

Question 4

Sol :

(i) $\left(x^{2}+3\right)(x-3)+9$

$x^{2}(x-3)+3(x-3)+9$

$x^{3}-3 x^{2}+3 x-9+9$

$x^{3}-3 x^{2}+3 x$

(ii) (x+3)(x-3)(x+4)(x-4)

[x(x-3)+3(x-3)][x(x-4)+4(x-4)]

$\left[x^{2}-3 x+3 x-9\right]\left[x^{2}-4 x+4 x-16\right]$

$\left[x^{2}-9\right]\left[x^{2}-16\right]$

$x^{2}\left(x^{2}-16\right)-9\left(x^{2}-16\right)$

$x^{4}-16 x^{2}-9 x^{2}+144$

$x^{4}-25 x^{4}+144$

(iii) (x+5)(x+6)(x+7)

[(x+5)(x+6)](x+7)

[x(x+6)+5(x+6)](x+7)

$\left(x^{2}+6 x+5 x+30\right)(x+7)$

$\left(x^{2}+6 x+5 x+30\right) x+\left(x^{v}+6 x+5 x+30\right) 7$

$\left(x^{2}+11 x+30\right) x+\left(x^{2}+11 x+30\right) 7$

$x^{3}+11 x^{2}+30 x+7 x^{2}+77 x+210$

$x^{3}+18 x^{2}+107 x+210$

(iv) (p+q-2 r)(2 p-q+r)-4 q r

p(2 p-q+r)+q(2 p-q+r)-2 r(2 p-q+r)-4 q r

$2 p^{2}-p q+p r+2 p q-q^{2}+q r-4 p y+2 q r-2 r^{2}-4 q r$

$2 p^{2}-q^{2}-2 r^{2}+p q-q r-3 p r$

(v) (p+q)(r+s)+(p-q)(r-s)-2(p r+q s) 

P(r+s)+q(r+s)+p(r-s)-q(r-s)-2 p r-2 q s

p r+p s+q r+q s+p r-p s-q r+q s-2 p r-2 q s

2 p r-2 p r+p s-p s+q r-q r+2 q s-2 q s

0+0+0+0=0

(vi) (x+y+z)(x-y+z)+(x+y-z)(-x+y+z)-4 z x

$x(x-y+z)+y(x-y+z)+z(x-y+z)+x\left(-x+y^{n}+z\right)$+y(-x+y+z)-z(-x+y+z)-4 z x

$x^{2}-x y+x z+x y-y^{v}+y z+x z-y z+z^{2}-x^{2}+x y+x z$ $-x y+y^{2}+y z+x z-y z-z^{2}-4 z x$

$x^{2}-x^{2}+2 x y-2 x y+4 x z-4 x z-y^{N}+y^{2}+2 y z-2 y z$ $+z^{2}-z^{2}$

0+0+0+0+0+0=0

Question 5

Sol :

Sides of rectangles $s_{1}=5 x^{2}+25 x y+4 y^{2}$

$S_{2}=2 x^{2}-2 x y+3 y r$

Area of rectangle  $=S_{1} \times S_{2}$

$A=\left(5 x^{2}+25 x y+4 y^{2}\right)\left(2 x^{2}-2 x y+3 y^{2}\right)$

$A=10 x^{4}+40 x^{3} y-27 x^{2} y^{2}+67 x y^{3}+12 y 4$