Exercise 12.1 (Revised) - Chapter 14 - Factorisation - Ncert Solutions class 8 - Maths

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Chapter 12: Factorisation - NCERT Solutions for Class 8 Maths

Ex 12.1 Question 1.

Find the common factors of the given terms.
(i) $12 x, 36$
(ii) $2 y, 22 x y$
(iii) $14 p q=28 p^2 q^2$
(iv) $2 x-3 x^2=4$
(v) $6 a b c, 24 a b^2=12 a^2 b$
(vi) $16 x^3=-4 x^2=32 x$
(vii) $10 p q=20 q r=30 r p$
(viii) $3 x^2 y^3=10 x^3 y^2=6 x^2 y^2 z$

Answer.

(i) $12 x=2 \times 2 \times 3 \times x$
$
36=2 \times 2 \times 3 \times 3
$

Hence, the common factors are 2,2 and $3=2 \times 2 \times 3=12$
(ii) $2 y=2 x y$
$
22 x y=2 \times 11 \times x \times y
$

Hence, the common factors are 2 and $y=2 \times y=2 y$
(iii) $14 p q=2 \times 7 \times p \times q$
$
28 p^2 q^2=2 \times 2 \times 7 \times p \times p \times q \times q
$

Hence, the common factors are $2 \times 7 \times p \times q=14 p q$
$
\begin{aligned}
& \text { (iv) } 2 x=2 \times x \times 1 \\
& 3 x^2=3 \times x \times x \times 1 \\
& 4=2 \times 2 \times 1
\end{aligned}
$

Hence, the common factor is 1 .
$
\begin{aligned}
& \text { (v) } 6 a b c=2 \times 3 \times a \times b \times c \\
& 24 a b^2=2 \times 2 \times 2 \times 3 \times a \times b \times b \\
& 12 a^2 b=2 \times 2 \times 3 \times a \times a \times b
\end{aligned}
$

Hence, the common factors are $2 \times 3 \times a \times b=6 a b$
$
\begin{aligned}
& \text { (vi) } 16 x^3=2 \times 2 \times 2 \times 2 \times x \times x \times x \\
& -4 x^2=(-1) \times 2 \times 2 \times x \times x \\
& 32 x=2 \times 2 \times 2 \times 2 \times 2 \times x
\end{aligned}
$

Hence the common factors are $2 \times 2 \times x=4 x$
(vii) $10 p q=2 \times 5 \times p \times q$
$
20 q r=2 \times 2 \times 5 \times q \times r
$

$
30 \mu p=2 \times 3 \times 5 \times r \times p
$

Hence the common factors are $2 \times 5=10$
$
\begin{aligned}
& \text { (viii) } 3 x^2 y^3=3 \times x \times x \times y \times y \times y \\
& 10 x^3 y^2=2 \times 5 \times x \times x \times x \times y \times y \\
& 6 x^2 y^2 z=2 \times 3 \times x \times x \times y \times y \times z
\end{aligned}
$

Hence the common factors are $x \times x \times y \times y=x^2 y^2$

Ex 12.1 Question 2.

Factorize the following expressions.
(i) $7 x-42$
(ii) $6 p-12 q$
(iii) $7 a^2+14 a$
(iv) $-16 z+20 z^3$
(v) $20 l^2 m+30 a l m$
(vi) $5 x^2 y-15 x y^2$
(vii) $10 a^2-15 b^2+20 c^2$
(viii) $-4 a^2+4 a b-4 c a$
(ix) $x^2 y z+x y^2 z+x y z^2$
(x) $a x^2 y+b x y^2+c x y z$

Answer.
(i) $7 x-42=7 \times x-2 \times 3 \times 7$

Taking common factors from each term,

$
\begin{aligned}
& =7(x-2 \times 3) \\
& =7(x-6)
\end{aligned}
$
(ii) $6 p-12 q=2 \times 3 \times p-2 \times 2 \times 3 \times q$

Taking common factors from each term,
$
\begin{aligned}
& =2 \times 3(p-2 q) \\
& =6(p-2 q)
\end{aligned}
$

(iii) $7 a^2+14 a=7 \times a \times a+2 \times 7 \times a$

Taking common factors from each term,
$
\begin{aligned}
& =7 \times a(a+2) \\
& =7 a(a+2)
\end{aligned}
$
(iv)
$
\begin{aligned}
& -16 z+20 z^3 \\
& =(-1) \times 2 \times 2 \times 2 \times 2 \times z+2 \times 2 \times 5 \times z \times z \times z
\end{aligned}
$

Taking common factors from each term,
$
\begin{aligned}
& =2 \times 2 \times z(-2 \times 2+5 \times z \times z) \\
& =4 z\left(-4+5 z^2\right)
\end{aligned}
$
(v)
$
\begin{aligned}
& 20 l^2 m+30 \mathrm{alm} \\
& =2 \times 2 \times 5 \times 1 \times 1 \times m+2 \times 3 \times 5 \times a \times 1 \times m
\end{aligned}
$

Taking common factors from each term,
$
=2 \times 5 \times l \times m(2 \times l+3 \times a)
$

$
=10 l m(2 l+3 a)
$
(vi) $5 x^2 y-15 x y^2=5 \times x \times x \times y+3 \times 5 \times x \times y \times y$ change the image with image_3310_1

Taking common factors from each term,
$
\begin{aligned}
& =5 \times x \times y(x-3 y) \\
& =5 x y(x-3 y)
\end{aligned}
$
(vii)

$
\begin{aligned}
& 10 a^2-15 b^2+20 c^2 \\
& =2 \times 5 \times a \times a-3 \times 5 \times b \times b+2 \times 2 \times 5 \times c \times c
\end{aligned}
$

Taking common factors from each term,
$
\begin{aligned}
& =5(2 \times a \times a-3 \times b \times b+2 \times 2 \times c \times c) \\
& =5\left(2 a^2-3 b^2+4 c^2\right)
\end{aligned}
$
(viii)
$
\begin{aligned}
& -4 a^2+4 a b-4 c a \\
& =(-1) \times 2 \times 2 \times a \times a+2 \times 2 \times a \times b-2 \times 2 \times c \times a
\end{aligned}
$

Taking common factors from each term,
$
\begin{aligned}
& =2 \times 2 \times a(-a+b-c) \\
& =4 a(-a+b-c)
\end{aligned}
$
(ix)
$
\begin{aligned}
& x^2 y z+x y^2 z+x y z^2 \\
& =x \times x \times y \times z+x \times y \times y \times z+x \times y \times z \times z
\end{aligned}
$

Taking common factors from each term,
$
=x \times y \times z(x+y+z)
$

(x)
$
\begin{aligned}
& a x^2 y+b x y^2+c o y z \\
& =a \times x \times x \times y+b \times x \times y \times y+c \times x \times y \times z
\end{aligned}
$

Taking common factors from each term,
$
\begin{aligned}
& =x \times y(a \times x+b \times y+c x z) \\
& =x y(a x+b y+c z)
\end{aligned}
$

Ex 12.1 Question 3.

Factorize:
(i) $x^2+x y+8 x+8 y$
(ii) $15 x y-6 x+5 y-2$
(iii) $a x+b x-a y-b y$
(iv) $15 p q+15+9 q+25 p$
(v) $z-7+7 x y-x y z$

Answer.

(i) $x^2+x y+8 x+8 y=x(x+y)+8(x+y)$
$
=(x+y)(x+8)
$
(ii) $15 x y-6 x+5 y-2=3 x(5 y-2)+1(5 y-2)$
$
=(5 y-2)(3 x+1)
$
(iii) $a x+b x-a y-b y=(a x+b x)-(a y+b y)=x(a+b)-y(a+b)$
$
=(a+b)(x-y)
$
(iv) $15 p q+15+9 q+25 p=15 p q+25 p+9 q+15$

$\begin{aligned}
&\begin{aligned}
& =5 p(3 q+5)+3(3 q+5) \\
& =(3 q+5)(5 p+3)
\end{aligned}\\
&\begin{aligned}
& \text { (v) } z-7+7 x y-x y z=7 x y-7-x y z+z \\
& =7(x y-1)-z(x y-1) \\
& =(x y-1)(7-z)=(-1)(1-x y)(-1)(z-7) \\
& =(1-x y)(z-7)
\end{aligned}
\end{aligned}$