NCERT Solutions for Class 8 Maths Chapter 14 Factorisation

Ans : Missing 

Ans : (i) \( 12 x=2 \times 2 \times 3 \times x  \)
\(36=2 \times 2 \times 3 \times 3   \)
The common factors are 2,2,3
And \(2 \times 2 \times 3=12   \) 


(ii) \(2 y=2 \times y   \)
\(  22 x y=2 \times 11 \times x \times y \)
The common Factors are 2,y
And \(  2 \times y=2 y \)


(iii) \(  14 p q=2 \times 7 \times p \times q \)
\( 28 p^{2} q^{2}=2 \times 2 \times 7 \times p \times p \times q \times q  \)
The common factors are \( 2,7, p, q \)
And \(2 \times 7 \times p \times q=14 p q   \)


(iv) \(2 x=2 \times x   \)
\(3 x^{2}=3 \times x \times x   \)
\(   4=2 \times 2\)
The common factor is 1


(v) \(6 a b c=2 \times 3 \times a \times b \times c   \)
\(24 a b^{2}=2 \times 2 \times 2 \times 3 \times a \times b \times b   \) 
\(12 a^{2} b=2 \times 2 \times 3 \times a \times a \times b   \) 
The common factors are \( 2,3, a, b  \) 
And \(  2 \times 3 \times a \times b=6 a b \) 


(vi) \(  3 x^{2} y^{3}=3 \times x \times x \times y \times y \times y \)
\(  -4 x^{2}=-1 \times 2 \times 2 \times x \times x \)
\( 32 x=2 \times 2 \times 2 \times 2 \times 2 \times x  \)
The common factors are  \(  2,2, x \)
And \( 2 \times 5=10  \)


(vii) \(10 p q=2 \times 5 \times p \times q   \) 
\(  20 q r=2 \times 2 \times 5 \times q \times r \)
\( 30 r p=2 \times 3 \times 5 \times r \times p  \)
The common factors are \(2,5   \)
And 2*5=10 


(viii) \(3 x^{2} y^{3}=3 \times x \times x \times y \times y \times y   \)
\( 10 x^{3} y^{2}=2 \times 5 \times x \times x \times x \times y \times y \)
\( 6 x^{2} y^{2} z=2 \times 3 \times x \times x \times y \times y \times z  \)
The common factors are \( x, x, y, y  \) 
And \( x \times x \times y \times y=x^{2} y^{2}  \) 

Ans : (i) \(7 x=7 \times x   \)
\( 42=2 \times 3 \times 7 \)
The common factor is 7 
\(  \therefore 7 x-42=(7 \times x)-(2 \times 3 \times 7)=7(x-6) \)


(ii) \( 6 p=2 \times 3 \times p  \)
The common factors are 2 and 3.
\( 12 q=2 \times 2 \times 3 \times q \)
\( \therefore 6 p-12 q=(2 \times 3 \times p)-(2 \times 2 \times 3 \times q)  \)
\( =2 \times 3[p-(2 \times q)]  \)
\(  =6(p-2 q) \) 


(iii) \( 14 a=2 \times 7 \times a  \)


\(\therefore 7 a^{2}+14 a=(7 \times a \times a)+(2 \times 7 \times a)   \)
\(  =7 \times a[a+2]=7 a(a+2) \)


(iv) \( 16 z=2 \times 2 \times 2 \times 2 \times z  \) 
\(  20 z^{3}=2 \times 2 \times 5 \times z \times z \times z \) 
The common factors \(2,2, \text { and } z  \)
\( \therefore-16 z+20 z^{3}=-(2 \times 2 \times 2 \times 2 \times z)+(2 \times 2 \times 5 \times z \times z \times z)  \) 
\(  =(2 \times 2 \times z)[-(2 \times 2)+(5 \times z \times z)] \) 
\(=4 z\left(-4+5 z^{2}\right)   \)


(v) \(  20 /^{2} m=2 \times 2 \times 5 \times / \times l \times m \)
\(  30 a / m=2 \times 3 \times 5 \times a \times l \times m \)
The common factors are 2,5,l and  m
\(  \therefore 20 /^{2} m+30 a / m=(2 \times 2 \times 5 \times I \times l \times m)+(2 \times 3 \times 5 \times a \times / \times m) \)
\( =(2 \times 5 \times I \times m)[(2 \times l)+(3 \times a)] \)
\(=10 \operatorName{lm}(2 l+3 a)   \)


(vi) \(  5 x^{2} y=5 \times x \times x \times y \)
\(15 x y^{2}=3 \times 5 \times x \times y \times y   \)
\(  =5 \times x \times y[x-(3 \times y)] \)
\(=5 x y(x-3 y)   \)
\(  =5 x y(x-3 y) \)


(vii) \( 10 a^{2}=2 \times 5 \times a \times a  \)
\( 15 b^{2}=3 \times 5 \times b \times b  \)
\(  20 c^{2}=2 \times 2 \times 5 \times c \times c \)
The common factor is 5.
\( 10 a^{2}-15 b^{2}+20 c^{2}=(2 \times 5 \times a \times a)-(3 \times 5 \times b \times b)+(2 \times 2 \times 5 \times c \times c)  \)
\(=5[(2 \times \mathrm{a} \times \mathrm{a})-(3 \times b \times b)+(2 \times 2 \times c \times c)]   \)
\( =5\left(2 a^{2}-3 b^{2}+4 c^{2}\right)  \)


(viii) \(4 a^{2}=2 \times 2 \times a \times a   \)
\( 4 a b=2 \times 2 \times a \times b \)
\(4 c a=2 \times 2 \times c \times a   \)
The common factors are 2,2 and a 
\(  \therefore-4 a^{2}+4 a b-4 c a=-(2 \times 2 \times a \times a)+(2 \times 2 \times a \times b)-(2 \times 2 \times c \times a) \)
\(  =2 \times 2 \times a[-(a)+b-c] \)
\( =4 a(-a+b-c)  \)


(ix) \(x^{2} y z=x \times x \times y \times z   \) 
\(  x y^{2} z=x \times y \times y \times z \)
\(x y z^{2}=x \times y \times z \times z   \)
The common factors are \(  x, y, \text { and } z \)
\( \therefore x^{2} y z+x y^{2} z+x y z^{2}=(x \times x \times y \times z)+(x \times y \times y \times z)+(x \times y \times z \times z)  \)
\(  =x \times y \times z[x+y+z] \)
\(  =x y z(x+y+z) \)


(x) \(  a x^{2} y=a \times x \times x \times y \)
\(  b x y^{2}=b \times x \times y \times y\)
\( c x y z=c \times x \times y \times z  \)
The common factors are x and y 
\(a x^{2} y+b x y^{2}+c x y z=(a \times x \times x \times y)+(b \times x \times y \times y)+(c \times x \times y \times z)   \)
\(  =(x \times y)[(a \times x)+(b \times y)+(c \times z)] \)
\( =x y(a x+b y+c z)  \) 

Ans : (i) \(x^{2}+x y+8 x+8 y=x \times x+x \times y+8 \times x+8 \times y   \)
\( =x(x+y)+8(x+y)  \)
\( =(x+y)(x+8)  \)


(ii) \(  15 x y-6 x+5 y-2=3 \times 5 \times x \times y-3 \times 2 \times x+5 \times y-2 \)
\(  =3 x(5 y-2)+1(5 y-2) \)
\( =(5 y-2)(3 x+1)  \)


(iii) \( a x+b x-a y-b y=a \times x+b \times x-a \times y-b \times y  \)
\(=x(a+b)-y(a+b)   \)
\( =(a+b)(x-y) \)


(iv) \(15 p q+15+9 q+25 p=15 p q+9 q+25 p+15   \)
\( =3 \times 5 \times p \times q+3 \times 3 \times q+5 \times 5 \times p+3 \times 5  \) 
\( =3 q(5 p+3)+5(5 p+3) \)


(v) \(  z-7+7 x y-x y z=z-x \times y \times z-7+7 \times x \times y \) 
\(=z(1-x y)-7(1-x y)   \)
\(=(1-x y)(z-7)   \) 

Ans : (i)  \(   a^{2}+8 a+16=(a)^{2}+2 \times a \times 4+(4)^{2}\)
\(  =(a+4)^{2}\left[(x+y)^{2}=x^{2}+2 x y+y^{2}\right] \)


(ii) \( p^{2}-10 p+25=(p)^{2}-2 \times p \times 5+(5)^{2}  \)
\( =(p-5)^{2}\left[(a-b)^{2}=a^{2}-2 a b+b^{2}\right]  \)


(iii) \( 25 m^{2}+30 m+9=(5 m)^{2}+2 \times 5 m \times 3+(3)^{2}  \)
\( =(5 m+3)^{2}\left[(a+b)^{2}=a^{2}+2 a b+b^{2}\right] \)


(iv) \( 49 y^{2}+84 y z+36 z^{2}=(7 y)^{2}+2 \times(7 y) \times(6 z)+(6 z)^{2} \)
\(  =(7 y+6 z)^{2}\left[(a+b)^{2}=a^{2}+2 a b+b^{2}\right] \)


(v) \( 4 x^{2}-8 x+4=(2 x)^{2}-2(2 x)(2)+(2)^{2}  \)
\( =(2 x-2)^{2}\left[(a-b)^{2}=a^{2}-2 a b+b^{2}\right]  \)
\( =[(2)(x-1)]^{2}=4(x-1)^{2}  \)


(vi) \(121 b^{2}-88 b c+16 c^{2}=(11 b)^{2}-2(11 b)(4 c)+(4 c)^{2}   \)
\( =(11 b-4 c)^{2}\left[(a-b)^{2}=a^{2}-2 a b+b^{2}\right]  \)


(v) \(  4 x^{2}-8 x+4=(2 x)^{2}-2(2 x)(2)+(2)^{2} \)
\(  =(2 x-2)^{2}\left[(a-b)^{2}=a^{2}-2 a b+b^{2}\right] \)
\( =[(2)(x-1)]^{2}=4(x-1)^{2}  \)


(vi) \( 121 b^{2}-88 b c+16 c^{2}=(11 b)^{2}-2(11 b)(4 c)+(4 c)^{2}  \)
\(  =(11 b-4 c)^{2}\left[(a-b)^{2}=a^{2}-2 a b+b^{2}\right] \)


(vii) \(  (I+m)^{2}-4 l m=f^{2}+2 l m+m^{2}-4 l m \)
\( =f^{2}-2 l m+m^{2}  \)


\(   =(I-m)^{2}\left[(a-b)^{2}=a^{2}-2 a b+b^{2}\right]\)


(viii) \( a^{4}+2 a^{2} b^{2}+b^{4}=\left(a^{2}\right)^{2}+2\left(a^{2}\right)\left(b^{2}\right)+\left(b^{2}\right)^{2}  \)
\(=\left(a^{2}+b^{2}\right)^{2}\left[(a+b)^{2}=a^{2}+2 a b+b^{2}\right]  \)