Exercise 8.2
Question 1
i)
$\begin{aligned} 7 x,-3 x & \\ 7 x-3 x &=4 x \end{aligned}$
ii) $6 x,-11 x$
$6 x-11 x=-5 x$
iii) $5 x^{2},-9 x^{2}$
$5 x^{2}-9 x^{2}=-4 x^{2}$
iv) $3 a b^{2},-5 a b^{2}$
$3 a b^{2}-5 a b^{2}=-4 a b^{2}$
v) $\frac{1}{2} p q,-1 / 3 p q$
$=\frac{1}{2} p q-\frac{1}{3} p q$
$=\frac{3 p q-2 p q}{3 \times 2}$
$=\frac{p q}{6}$
vi)$5 x^{3} y,-\frac{2}{3} x^{3} y$
$=5 x^{3} y-\frac{2}{3} x^{3} y$
$=\frac{15 x^{3} y-2 x^{3} y}{3}$
$=\frac{13 x^{3} y}{3}$
Question 2
i)
$\begin{aligned} 3 x-5 x, 7 x & \\ 3 x-5 x+7 x=5 x \end{aligned}$
ii) 7xy, $2 x y$, $-8 x y$
=$27 x y+2 x y-8 x y$
=xy
iii) $-2 a b c, 3 a b c, a b c$
$-2 a b c+3 a b c+a b c=2 a b c$
iv) $3 m n,-5 m n, 8 m n,-4 m n$
$3 m n-5 m n+8 m n-4 m n=2 m n$
v) $2 x^{3}, 3 x^{3},-4 x^{3},-5 x^{3}$
$2 x^{3}+3 x^{3}-4 x^{3}-5 x^{3}$
$5 x^{3}-9 x^{3}=-4 x^{3}$
Question 3
i) $8 b-32$
ii) $8 m^{2}-11 m+10$
iii) $7 z^{3}+12 z^{2}-202$
iv) $8 x^{2} y+$ 8xy $^{2}-4 x^{2}-7 y^{2}$
v) p-q
vi) $\quad a+a b$
vii) $ \quad 4 y^{2}-3 y$
Question 4
i) $5 x y,-7 x y, 3 x^{2}$
=$5 x y-7 x y+3 x^{2}$
=$3 x^{2}-2 x y$
ii) $4 x^{2} y_{2}-3 x y^{2},-8 x y^{2}, 5 x^{2} y$
=$4 x^{2} y-3 x y^{2}-5 x y^{2}+5 x^{2} y$
=$9 x^{2} y-8 x y^{2}$
iii) $-7 m n+5,12 m n+12,8 m n-8,-2 m n-3$
=$-7 m n+5+18 m n+12+8 m n-8-2 m n-3$
=$11 m n-4$
iv) $a+b-3, \quad b-a+3, a-b+3$
=$a+b-3+b-a+3+a-b+3$
=$\quad a+b+3$
v) $14 x+10 y-12 x y-13, \quad 18-7 x-10 y+8 x y, 4 x y$
=$14 x+10 y-12 x y-13+18-7 x-10 y+8 x y+4 x y$
=$7 x+5$
vi) $5 m-7 n, \quad 3 n-4 m+2,2 m-3 m n-5$
=$5 m-7 n+3 n-4 m+2+2 m-3 m n-5$
=$3 m-4 n-3 m n-3$
vii)$7 a^{2}-5 a+2, \quad 3 a^{2}-7, \quad 2 a+a, \quad 1+2 a-5 a^{2}$
=$7 a^{2}-5 a+2+3 a^{2}-7+2 a+9+1+2 a-5 a^{2}$
=$5 a^{2}-a+5$
Question 5
i) $2 x^{2}+3 y^{2}-5 x y+5 x^{2}-y^{2}+6 x y-3 x^{2}$
=$x^{2}(2+3)+y^{2}(3-1)+x y(-5+6)$
=$4 x^{2}+2 y^{2}+x y$
ii) $3 x y^{2}-5 x^{2} y+7 x y-8 y^{2} x-4 x y+6 x^{2} y$
=$x y^{2}(3-8)+x^{2} y(-5+6)+x y(7-4)$
=$-5 x y^{2}+6 x^{2} y+3 x y$
iii) $5 x^{4}-7 x^{2}+8 x-1+3 x^{3}-9 x^{2}+7-3 x^{4}+11 x-2+8 x^{2}$
=$x^{4}(5-3)+x^{2}(-7-9+8)+x(8+11)+3 x^{3}-1+7-2$
=$2 x^{4}-8 x^{2}+19 x+3 x^{3}+4$
=$2 x^{4}+3 x^{3}-8 x^{2}+19 x+4$
Question 6
i) $y^{2}-\left(-5 y^{2}\right)$
$y^{2}+5 y^{2}=6 y^{2}$
ii) $-2 x y-(-7 x y)$
$-2 x y+7 x y$
$5 x y$
iii) $b(5-a)-a(b-5)$
$5 b-a b-a b+5 a$
$5 a+5 b-2 a b$
iv) $4 m^{2}-3 m n+8-\left(-m^{2}+5 m n\right)$
$4 m^{2}-3 m n+8+m^{2}-5 m n$
$5 m^{2}-8 m n+8$
v)$3 a b-2 a^{2}-2 b^{2}-\left(5 a^{2}-7 a b+5 b^{2}\right)$
$3 a b-2 a^{2}-2 b^{2}-5 a^{2}+7 a b-5 b^{2}$
$10 a b-7 a^{2}-7 b^{2}$
vi) $5 p^{2}+3 q^{2}-p q-\left(4 p q-5 q^{2}-3 p^{2}\right)$
$5 p^{2}+3 q^{2}-p q-4 p q+5 q^{2}+3 p^{2}$
$8 p^{2}+8 q^{2}-5 p q$
vii) $7 x^{2}-8 x y+3 y^{2}-5-\left(7 x y+5 x^{2}-7 y^{2}+3\right)$
$7 x^{2}-8 x y+3 y^{2}-5-7 x y-5 x^{2}+7 y^{2}-3$
$x^{2}(7-5)+x y(-8-7)+y^{2}(3+7)-8$
$2 x^{2}-15 x y+10 y^{2}-8$
viii) $\left(x^{4}-3 x^{3}-2 x^{2}+3\right)-\left(2 x^{4}-7 x^{2}+5 x+3\right)$
$x^{4}-3 x^{3}-2 x^{2}+3-2 x^{4}+7 x^{2}-5 x-3$
$x^{4}(1-2)-3 x^{3}+x^{2}(-2+7)-5 x$
$-x^{4}-3 x^{3}+5 x^{2}-5 x$
Question 7
Sum of ( 10p - r ) and 5P+ 2q is
$=10 p- r+5 p+2 q$
=$15 p+2 q-r$
$P-2 q+r$ Subtract from $15 p+2 q-r$
=$(15 p+2 q-r)-(p-2 q+r)$
=$15 p+2 q-r-p+2 q-r$
=$p(15-1)+q(2+2)+r(-1-1)$
=$14 p+4 q-2 r$
Question 8
Sum of $4+3 x$ and $5-4 x+2 x^{2}$
$2 x^{2}-4 x+5+4+3 x$
$2 x^{2}-x+9$
Sum of $3 x^{2}-5 x$ and $-x^{2}+2 x+5$
=$3 x^{2}-5 x+\left(-x^{2}+2 x+5\right)$
=$2 x^{2}-3 x+5$
$2 x^{2}-x+9$ is substracted from $2 x^{2}-3 x+5$
=$2 x^{2}-3 x+5-\left(2 x^{2}-x+9\right)$
=$2 x^{2}-3 x+5-2 x^{2}+x-9$
=$2x-4$
Question 9
Sum is $x^{2}+y^{2}+5 x y$
Subtract $x^{2}-y^{2}+2 x y$ from $x^{2}+y^{2}+5 x y$
$x^{2}+y^{2}+5 x y$
$x^{2}-y^{2}+2 x y$
$-$ $+$ $-$
---------------------------------
$2 y^{2}+3 x y$
Question 10
$-7 m n+2 m^{2}+3 n^{2}$
$2 m n+m^{2}+n^{2}$
$-$ $-$ $-$
------------------------------------
$-9 m n+m^{2}+2 n^{2}$
Question 11
The required
$y^{4}-12 y^{2}+y+14-\left(17 y^{3}+34 y^{2}-51 y+68\right)$
$y^{4}-17 y^{3}+y^{2}(-12-34)+y(1+51)+14-68$
$y^{4}-17 y^{3}-46 y^{2}+52 y-54$
Question 12
The required
$93 p^{2}-55 p+4-\left(13 p^{3}-5 p^{2}+17 p-90\right)$
$93 p^{2}-55 p+4-13 p^{3}+5 p^{2}-17 p+90$
$-13 p^{3}+98 p^{2}-72 p+94$
Question 13
The required expressions
$3 x^{2}-4 y^{2}+5 x y+20$
$-x^{2}-y^{2}+6 x y+20$
$+$ $+$ $-$ $-$
-------------------------------------
$4 x^{2}-2 y^{2}-x y$
Question 14
Sum of $2 y^{2}+3 y z,-y^{2}-y z-z^{2}, \quad y z+2 z^{2}$ is
$2 y^{2}+3 y z-y^{2}-y z-z^{2}+y z+2 z^{2}$
$y^{2}(2-1) 7 y z(3-1+1)+z^{2}(-1+2)$
$y^{2}+3 y z+z^{2}$
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