Class 8 Exam > Class 8 Notes > Mathematics (Maths) Class 8 > RS Aggarwal Solutions: Exercise 7A - Factorisation
RS Aggarwal Solutions: Exercise 7A - Factorisation | Mathematics (Maths) Class 8 PDF Download
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Page 1
Q u e s t i o n : 1
Factorise:
i 12x + 15
ii 14m - 21
iii 9n - 12n
2
S o l u t i o n :
i 12x +15 = 3(4x +5)
ii 14m -21 = 7(2m -3)
iii 9n -12n
2
= 3n(3 -4n)
Q u e s t i o n : 2
Factorise:
i 16a
2
- 24ab
ii 15ab
2
- 20a
2
b
iii 12x
2
y
3
- 21x
3
y
2
S o l u t i o n :
i H.C.F. of 16a
2
and 24ab is 8a.
? 16a
2
-24ab = 8a(2a -3b)
ii H.C.F. of 15ab
2
and 20a
2
b is 5ab.
? 15ab
2
-20a
2
b = 5ab(3b -4a)
iii ?H.C.F. of 12x
2
y
3
and 21x
3
y
2
is 3x
2
y
2
.
? 12x
2
y
3
-21x
3
y
2
= 3x
2
y
2
(4y -7x)
Q u e s t i o n : 3
Factorise:
i 24x
3
- 36x
2
y
ii 10x
3
- 15x
2
iii 36x
3
y - 60x
2
y
3
z
S o l u t i o n :
i H.C.F. of 24x
3
and 36x
2
y is 6x
2
.
? 24x
3
-36x
2
y = 6x
2
(4x -6y)
ii H.C.F. of 10x
3
and 15x
2
is 5x
3
.
? 10x
3
-15x
2
= 5x
2
(2x -3)
iii H.C.F. of 36x
3
y and 60x
2
y
3
z is 12x
2
y.
? 36x
3
y -60x
2
y
3
z = 12x
2
y 3x -5y
2
z
Q u e s t i o n : 4
Factorise:
i 9x
3
- 6x
2
+ 12x
ii 8x
2
- 72xy + 12x
iii 18a
3
b
3
- 27a
2
b
3
+ 36a
3
b
2
S o l u t i o n :
i H.C.F. of 9x
3
, 6x
2
and 12x is 3x.
? 9x
3
-6x
2
+12x = 3x 3x
2
-2x +4
ii H.C.F. of 8x
3
, 72xy and 12x is 4x.
? 8x
3
-72xy +12x = 4x 2x
2
-18y +3
iii H.C.F. of 18a
3
b
3
, 27a
2
b
3
and 36a
3
b
2
is 9a
2
b
2
.
? 18a
3
b
3
-27a
2
b
3
+36a
3
b
2
= 9a
2
b
2
(2ab -3b +4a)
Q u e s t i o n : 5
Factorise:
i 14x
3
+ 21x
4
y - 28x
2
y
2
ii -5 - 10t + 20t
2
S o l u t i o n :
i H.C.F. of 14x
3
, 21x
4
y and 28x
2
y
2
is 7x
2
.
? 14x
3
+21x
4
y -28x
2
y
2
= 7x
2
2x +3x
2
y -4y
2
ii H.C.F. of -5, -10t and 20t
2
is 5.
? -5 -10t +20t
2
= 5 -1 -2t +4t
2
Q u e s t i o n : 6
Factorise:
( )
( )
( )
( )
( )
Page 2
Q u e s t i o n : 1
Factorise:
i 12x + 15
ii 14m - 21
iii 9n - 12n
2
S o l u t i o n :
i 12x +15 = 3(4x +5)
ii 14m -21 = 7(2m -3)
iii 9n -12n
2
= 3n(3 -4n)
Q u e s t i o n : 2
Factorise:
i 16a
2
- 24ab
ii 15ab
2
- 20a
2
b
iii 12x
2
y
3
- 21x
3
y
2
S o l u t i o n :
i H.C.F. of 16a
2
and 24ab is 8a.
? 16a
2
-24ab = 8a(2a -3b)
ii H.C.F. of 15ab
2
and 20a
2
b is 5ab.
? 15ab
2
-20a
2
b = 5ab(3b -4a)
iii ?H.C.F. of 12x
2
y
3
and 21x
3
y
2
is 3x
2
y
2
.
? 12x
2
y
3
-21x
3
y
2
= 3x
2
y
2
(4y -7x)
Q u e s t i o n : 3
Factorise:
i 24x
3
- 36x
2
y
ii 10x
3
- 15x
2
iii 36x
3
y - 60x
2
y
3
z
S o l u t i o n :
i H.C.F. of 24x
3
and 36x
2
y is 6x
2
.
? 24x
3
-36x
2
y = 6x
2
(4x -6y)
ii H.C.F. of 10x
3
and 15x
2
is 5x
3
.
? 10x
3
-15x
2
= 5x
2
(2x -3)
iii H.C.F. of 36x
3
y and 60x
2
y
3
z is 12x
2
y.
? 36x
3
y -60x
2
y
3
z = 12x
2
y 3x -5y
2
z
Q u e s t i o n : 4
Factorise:
i 9x
3
- 6x
2
+ 12x
ii 8x
2
- 72xy + 12x
iii 18a
3
b
3
- 27a
2
b
3
+ 36a
3
b
2
S o l u t i o n :
i H.C.F. of 9x
3
, 6x
2
and 12x is 3x.
? 9x
3
-6x
2
+12x = 3x 3x
2
-2x +4
ii H.C.F. of 8x
3
, 72xy and 12x is 4x.
? 8x
3
-72xy +12x = 4x 2x
2
-18y +3
iii H.C.F. of 18a
3
b
3
, 27a
2
b
3
and 36a
3
b
2
is 9a
2
b
2
.
? 18a
3
b
3
-27a
2
b
3
+36a
3
b
2
= 9a
2
b
2
(2ab -3b +4a)
Q u e s t i o n : 5
Factorise:
i 14x
3
+ 21x
4
y - 28x
2
y
2
ii -5 - 10t + 20t
2
S o l u t i o n :
i H.C.F. of 14x
3
, 21x
4
y and 28x
2
y
2
is 7x
2
.
? 14x
3
+21x
4
y -28x
2
y
2
= 7x
2
2x +3x
2
y -4y
2
ii H.C.F. of -5, -10t and 20t
2
is 5.
? -5 -10t +20t
2
= 5 -1 -2t +4t
2
Q u e s t i o n : 6
Factorise:
( )
( )
( )
( )
( )
i x(x + 3) + 5(x + 3)
ii 5x(x - 4) - 7(x - 4)
iii 2m(1 - n) + 3(1 - n)
S o l u t i o n :
i x(x +3)+5(x +3) = (x +3)(x +5)
ii 5x(x -4)-7(x -4) = (x -4)(5x -7)
iii 2m(1 -n)+3(1 -n) = (1 -n)(2m +3)
Q u e s t i o n : 7
Factorise:
6a(a - 2b) + 5b(a - 2b)
S o l u t i o n :
We have:
6a(a -2b)+5b(a -2b) = (a -2b)(6a +5b)
Q u e s t i o n : 8
Factorise:
x
3
(2a - b) + x
2
(2a - b)
S o l u t i o n :
We have:
x
3
(2a -b)+x
2
(2a -b) = (2a -b) x
3
+x
2
= x
2
(x +1)(2a -b)
Q u e s t i o n : 9
Factorise:
9a(3a - 5b) - 12a
2
(3a - 5b)
S o l u t i o n :
We have:
9a(3a -5b)-12a
2
(3a -5b) = (3a -5b) 9a -12a
2
= 3a(3a -5b)(3 -4a)
Q u e s t i o n : 1 0
Factorise:
(x + 5)
2
- 4(x + 5)
S o l u t i o n :
We have:
(x +5)
2
-4(x +5) = (x +5){(x +5)-4}
= (x +5)(x +5 -4) = (x +5)(x +1)
? (x +5)
2
-4(x +5) = (x +5)(x +1)
Q u e s t i o n : 1 1
Factorise:
3(a - 2b)
2
- 5(a - 2b)
S o l u t i o n :
3(a -2b)
2
-5(a -2b) = (a -2b){3(a -2b)-5}
= (a -2b)(3a -6b -5)
? 3(a -2b)
2
-5(a -2b) = (a -2b)(3a -6b -5)
Q u e s t i o n : 1 2
Factorise:
2a + 6b - 3(a + 3b)
2
S o l u t i o n :
We have:
2a +6b -3(a +3b)
2
= 2(a +3b)-3(a +3b)
2
= (a +3b){2 -3(a +3b)} = (a +3b)(2 -3a -9b)
? 2a +6b -3(a +3b)
2
= (a +3b)(2 -3a -9b)
Q u e s t i o n : 1 3
Factorise:
16(2p - 3q)
2
- 4(2p - 3q)
S o l u t i o n :
We have:
16(2p -3q)
2
-4(2p -3q) = (2p -3q){16(2p -3q)-4} = (2p -3q)(32p -48q -4)
? 16(2p -3q)
2
-4(2p -3q) = (2p -3q)(32p -48q -4)
Q u e s t i o n : 1 4
Factorise:
x(a - 3) + y(3 - a)
S o l u t i o n :
We have:
x(a -3)+y(3 -a) = x(a -3)-y(a -3) = (a -3)(x -y)
? x(a -3)+y(3 -a) = (a -3)(x -y)
Q u e s t i o n : 1 5
Factorise:
12(2x - 3y)
2
- 16(3y - 2x)
( )
( )
Page 3
Q u e s t i o n : 1
Factorise:
i 12x + 15
ii 14m - 21
iii 9n - 12n
2
S o l u t i o n :
i 12x +15 = 3(4x +5)
ii 14m -21 = 7(2m -3)
iii 9n -12n
2
= 3n(3 -4n)
Q u e s t i o n : 2
Factorise:
i 16a
2
- 24ab
ii 15ab
2
- 20a
2
b
iii 12x
2
y
3
- 21x
3
y
2
S o l u t i o n :
i H.C.F. of 16a
2
and 24ab is 8a.
? 16a
2
-24ab = 8a(2a -3b)
ii H.C.F. of 15ab
2
and 20a
2
b is 5ab.
? 15ab
2
-20a
2
b = 5ab(3b -4a)
iii ?H.C.F. of 12x
2
y
3
and 21x
3
y
2
is 3x
2
y
2
.
? 12x
2
y
3
-21x
3
y
2
= 3x
2
y
2
(4y -7x)
Q u e s t i o n : 3
Factorise:
i 24x
3
- 36x
2
y
ii 10x
3
- 15x
2
iii 36x
3
y - 60x
2
y
3
z
S o l u t i o n :
i H.C.F. of 24x
3
and 36x
2
y is 6x
2
.
? 24x
3
-36x
2
y = 6x
2
(4x -6y)
ii H.C.F. of 10x
3
and 15x
2
is 5x
3
.
? 10x
3
-15x
2
= 5x
2
(2x -3)
iii H.C.F. of 36x
3
y and 60x
2
y
3
z is 12x
2
y.
? 36x
3
y -60x
2
y
3
z = 12x
2
y 3x -5y
2
z
Q u e s t i o n : 4
Factorise:
i 9x
3
- 6x
2
+ 12x
ii 8x
2
- 72xy + 12x
iii 18a
3
b
3
- 27a
2
b
3
+ 36a
3
b
2
S o l u t i o n :
i H.C.F. of 9x
3
, 6x
2
and 12x is 3x.
? 9x
3
-6x
2
+12x = 3x 3x
2
-2x +4
ii H.C.F. of 8x
3
, 72xy and 12x is 4x.
? 8x
3
-72xy +12x = 4x 2x
2
-18y +3
iii H.C.F. of 18a
3
b
3
, 27a
2
b
3
and 36a
3
b
2
is 9a
2
b
2
.
? 18a
3
b
3
-27a
2
b
3
+36a
3
b
2
= 9a
2
b
2
(2ab -3b +4a)
Q u e s t i o n : 5
Factorise:
i 14x
3
+ 21x
4
y - 28x
2
y
2
ii -5 - 10t + 20t
2
S o l u t i o n :
i H.C.F. of 14x
3
, 21x
4
y and 28x
2
y
2
is 7x
2
.
? 14x
3
+21x
4
y -28x
2
y
2
= 7x
2
2x +3x
2
y -4y
2
ii H.C.F. of -5, -10t and 20t
2
is 5.
? -5 -10t +20t
2
= 5 -1 -2t +4t
2
Q u e s t i o n : 6
Factorise:
( )
( )
( )
( )
( )
i x(x + 3) + 5(x + 3)
ii 5x(x - 4) - 7(x - 4)
iii 2m(1 - n) + 3(1 - n)
S o l u t i o n :
i x(x +3)+5(x +3) = (x +3)(x +5)
ii 5x(x -4)-7(x -4) = (x -4)(5x -7)
iii 2m(1 -n)+3(1 -n) = (1 -n)(2m +3)
Q u e s t i o n : 7
Factorise:
6a(a - 2b) + 5b(a - 2b)
S o l u t i o n :
We have:
6a(a -2b)+5b(a -2b) = (a -2b)(6a +5b)
Q u e s t i o n : 8
Factorise:
x
3
(2a - b) + x
2
(2a - b)
S o l u t i o n :
We have:
x
3
(2a -b)+x
2
(2a -b) = (2a -b) x
3
+x
2
= x
2
(x +1)(2a -b)
Q u e s t i o n : 9
Factorise:
9a(3a - 5b) - 12a
2
(3a - 5b)
S o l u t i o n :
We have:
9a(3a -5b)-12a
2
(3a -5b) = (3a -5b) 9a -12a
2
= 3a(3a -5b)(3 -4a)
Q u e s t i o n : 1 0
Factorise:
(x + 5)
2
- 4(x + 5)
S o l u t i o n :
We have:
(x +5)
2
-4(x +5) = (x +5){(x +5)-4}
= (x +5)(x +5 -4) = (x +5)(x +1)
? (x +5)
2
-4(x +5) = (x +5)(x +1)
Q u e s t i o n : 1 1
Factorise:
3(a - 2b)
2
- 5(a - 2b)
S o l u t i o n :
3(a -2b)
2
-5(a -2b) = (a -2b){3(a -2b)-5}
= (a -2b)(3a -6b -5)
? 3(a -2b)
2
-5(a -2b) = (a -2b)(3a -6b -5)
Q u e s t i o n : 1 2
Factorise:
2a + 6b - 3(a + 3b)
2
S o l u t i o n :
We have:
2a +6b -3(a +3b)
2
= 2(a +3b)-3(a +3b)
2
= (a +3b){2 -3(a +3b)} = (a +3b)(2 -3a -9b)
? 2a +6b -3(a +3b)
2
= (a +3b)(2 -3a -9b)
Q u e s t i o n : 1 3
Factorise:
16(2p - 3q)
2
- 4(2p - 3q)
S o l u t i o n :
We have:
16(2p -3q)
2
-4(2p -3q) = (2p -3q){16(2p -3q)-4} = (2p -3q)(32p -48q -4)
? 16(2p -3q)
2
-4(2p -3q) = (2p -3q)(32p -48q -4)
Q u e s t i o n : 1 4
Factorise:
x(a - 3) + y(3 - a)
S o l u t i o n :
We have:
x(a -3)+y(3 -a) = x(a -3)-y(a -3) = (a -3)(x -y)
? x(a -3)+y(3 -a) = (a -3)(x -y)
Q u e s t i o n : 1 5
Factorise:
12(2x - 3y)
2
- 16(3y - 2x)
( )
( )
S o l u t i o n :
We have:
12(2x -3y)
2
-16(3y -2x) = 12(2x -3y)
2
+16(2x -3y) = (2x -3y){12(2x -3y)+16} = (2x -3y)(24x -36y +16)
? 12(2x -3y)
2
-16(3y -2x) = (2x -3y)(24x -36y +16)
Q u e s t i o n : 1 6
Factorise:
(x + y)(2x + 5) - (x + y)(x + 3)
S o l u t i o n :
We have:
(x +y)(2x +5)-(x +y)(x +3) = (x +y){(2x +5)-(x +3)} = (x +y)(2x +5 -x -3) = (x +y)(x +2)
Q u e s t i o n : 1 7
Factorise:
ar + br + at + bt
S o l u t i o n :
By grouping the terms:
ar+br+at +bt = (ar+br)+(at +bt) = r(a +b)+t(a +b) = (a +b)(r+t)
? ar+br+at +bt = (a +b)(r+t)
Q u e s t i o n : 1 8
Factorise:
x
2
- ax - bx + ab
S o l u t i o n :
By suitably arranging the terms:
x
2
-ax -bx +ab = x
2
-bx -ax +ab = x
2
-bx -(ax -ab) = x(x -b)-a(x -b) = (x -b)(x -a)
? x
2
-ax -bx +ab = (x -b)(x -a)
Q u e s t i o n : 1 9
Factorise:
ab
2
- bc
2
- ab + c
2
S o l u t i o n :
By suitably arranging the terms:
ab
2
-bc
2
-ab +c
2
= ab
2
-ab -bc
2
+c
2
= ab
2
-ab - bc
2
-c
2
= ab(b -1)-c
2
(b -1) = (b -1) ab -c
2
? ab
2
-bc
2
-ab +c
2
= (b -1) ab -c
2
Q u e s t i o n : 2 0
Factorise:
x
2
- xz + xy - yz
S o l u t i o n :
By suitably arranging the terms:
x
2
-xz +xy -yz = x
2
+xy -xz -yz = x
2
+xy -(xz +yz) = x(x +y)-z(x +y) = (x +y)(x -z)
? x
2
-xz +xy -yz = (x +y)(x -z)
Q u e s t i o n : 2 1
Factorise:
6ab - b
2
+ 12ac - 2bc
S o l u t i o n :
By suitably arranging the terms:
6ab -b
2
+12ac -2bc = 6ab +12ac -b
2
-2bc = (6ab +12ac)- b
2
+2bc = 6a(b +2c)-b(b +2c) = (b +2c)(6a -b)
? 6ab -b
2
+12ac -2bc = (b +2c)(6a -b)
Q u e s t i o n : 2 2
Factorise:
(x - 2y)
2
+ 4x - 8y
S o l u t i o n :
We have:
(x -2y)
2
+4x -8y = (x -2y)
2
+4(x -2y) = (x -2y)(x -2y)+4(x -2y) = (x -2y){(x -2y)+4} = (x -2y)(x -2y +4)
? (x -2y)
2
+4x -8y = (x -2y)(x -2y +4)
Q u e s t i o n : 2 3
Factorise:
y
2
- xy(1 - x) - x
3
S o l u t i o n :
We have:
y
2
-xy(1 -x)-x
3
= y
2
-xy +x
2
y -x
3
= y
2
-xy + x
2
y -x
3
= y(y -x)+x
2
(y -x) = (y -x) y +x
2
? y
2
-xy(1 -x)-x
3
= (y -x) y +x
2
Q u e s t i o n : 2 4
Factorise:
( )
( ) ( ) ( )
( )
( )
( )
( ) ( ) ( )
( )
Page 4
Q u e s t i o n : 1
Factorise:
i 12x + 15
ii 14m - 21
iii 9n - 12n
2
S o l u t i o n :
i 12x +15 = 3(4x +5)
ii 14m -21 = 7(2m -3)
iii 9n -12n
2
= 3n(3 -4n)
Q u e s t i o n : 2
Factorise:
i 16a
2
- 24ab
ii 15ab
2
- 20a
2
b
iii 12x
2
y
3
- 21x
3
y
2
S o l u t i o n :
i H.C.F. of 16a
2
and 24ab is 8a.
? 16a
2
-24ab = 8a(2a -3b)
ii H.C.F. of 15ab
2
and 20a
2
b is 5ab.
? 15ab
2
-20a
2
b = 5ab(3b -4a)
iii ?H.C.F. of 12x
2
y
3
and 21x
3
y
2
is 3x
2
y
2
.
? 12x
2
y
3
-21x
3
y
2
= 3x
2
y
2
(4y -7x)
Q u e s t i o n : 3
Factorise:
i 24x
3
- 36x
2
y
ii 10x
3
- 15x
2
iii 36x
3
y - 60x
2
y
3
z
S o l u t i o n :
i H.C.F. of 24x
3
and 36x
2
y is 6x
2
.
? 24x
3
-36x
2
y = 6x
2
(4x -6y)
ii H.C.F. of 10x
3
and 15x
2
is 5x
3
.
? 10x
3
-15x
2
= 5x
2
(2x -3)
iii H.C.F. of 36x
3
y and 60x
2
y
3
z is 12x
2
y.
? 36x
3
y -60x
2
y
3
z = 12x
2
y 3x -5y
2
z
Q u e s t i o n : 4
Factorise:
i 9x
3
- 6x
2
+ 12x
ii 8x
2
- 72xy + 12x
iii 18a
3
b
3
- 27a
2
b
3
+ 36a
3
b
2
S o l u t i o n :
i H.C.F. of 9x
3
, 6x
2
and 12x is 3x.
? 9x
3
-6x
2
+12x = 3x 3x
2
-2x +4
ii H.C.F. of 8x
3
, 72xy and 12x is 4x.
? 8x
3
-72xy +12x = 4x 2x
2
-18y +3
iii H.C.F. of 18a
3
b
3
, 27a
2
b
3
and 36a
3
b
2
is 9a
2
b
2
.
? 18a
3
b
3
-27a
2
b
3
+36a
3
b
2
= 9a
2
b
2
(2ab -3b +4a)
Q u e s t i o n : 5
Factorise:
i 14x
3
+ 21x
4
y - 28x
2
y
2
ii -5 - 10t + 20t
2
S o l u t i o n :
i H.C.F. of 14x
3
, 21x
4
y and 28x
2
y
2
is 7x
2
.
? 14x
3
+21x
4
y -28x
2
y
2
= 7x
2
2x +3x
2
y -4y
2
ii H.C.F. of -5, -10t and 20t
2
is 5.
? -5 -10t +20t
2
= 5 -1 -2t +4t
2
Q u e s t i o n : 6
Factorise:
( )
( )
( )
( )
( )
i x(x + 3) + 5(x + 3)
ii 5x(x - 4) - 7(x - 4)
iii 2m(1 - n) + 3(1 - n)
S o l u t i o n :
i x(x +3)+5(x +3) = (x +3)(x +5)
ii 5x(x -4)-7(x -4) = (x -4)(5x -7)
iii 2m(1 -n)+3(1 -n) = (1 -n)(2m +3)
Q u e s t i o n : 7
Factorise:
6a(a - 2b) + 5b(a - 2b)
S o l u t i o n :
We have:
6a(a -2b)+5b(a -2b) = (a -2b)(6a +5b)
Q u e s t i o n : 8
Factorise:
x
3
(2a - b) + x
2
(2a - b)
S o l u t i o n :
We have:
x
3
(2a -b)+x
2
(2a -b) = (2a -b) x
3
+x
2
= x
2
(x +1)(2a -b)
Q u e s t i o n : 9
Factorise:
9a(3a - 5b) - 12a
2
(3a - 5b)
S o l u t i o n :
We have:
9a(3a -5b)-12a
2
(3a -5b) = (3a -5b) 9a -12a
2
= 3a(3a -5b)(3 -4a)
Q u e s t i o n : 1 0
Factorise:
(x + 5)
2
- 4(x + 5)
S o l u t i o n :
We have:
(x +5)
2
-4(x +5) = (x +5){(x +5)-4}
= (x +5)(x +5 -4) = (x +5)(x +1)
? (x +5)
2
-4(x +5) = (x +5)(x +1)
Q u e s t i o n : 1 1
Factorise:
3(a - 2b)
2
- 5(a - 2b)
S o l u t i o n :
3(a -2b)
2
-5(a -2b) = (a -2b){3(a -2b)-5}
= (a -2b)(3a -6b -5)
? 3(a -2b)
2
-5(a -2b) = (a -2b)(3a -6b -5)
Q u e s t i o n : 1 2
Factorise:
2a + 6b - 3(a + 3b)
2
S o l u t i o n :
We have:
2a +6b -3(a +3b)
2
= 2(a +3b)-3(a +3b)
2
= (a +3b){2 -3(a +3b)} = (a +3b)(2 -3a -9b)
? 2a +6b -3(a +3b)
2
= (a +3b)(2 -3a -9b)
Q u e s t i o n : 1 3
Factorise:
16(2p - 3q)
2
- 4(2p - 3q)
S o l u t i o n :
We have:
16(2p -3q)
2
-4(2p -3q) = (2p -3q){16(2p -3q)-4} = (2p -3q)(32p -48q -4)
? 16(2p -3q)
2
-4(2p -3q) = (2p -3q)(32p -48q -4)
Q u e s t i o n : 1 4
Factorise:
x(a - 3) + y(3 - a)
S o l u t i o n :
We have:
x(a -3)+y(3 -a) = x(a -3)-y(a -3) = (a -3)(x -y)
? x(a -3)+y(3 -a) = (a -3)(x -y)
Q u e s t i o n : 1 5
Factorise:
12(2x - 3y)
2
- 16(3y - 2x)
( )
( )
S o l u t i o n :
We have:
12(2x -3y)
2
-16(3y -2x) = 12(2x -3y)
2
+16(2x -3y) = (2x -3y){12(2x -3y)+16} = (2x -3y)(24x -36y +16)
? 12(2x -3y)
2
-16(3y -2x) = (2x -3y)(24x -36y +16)
Q u e s t i o n : 1 6
Factorise:
(x + y)(2x + 5) - (x + y)(x + 3)
S o l u t i o n :
We have:
(x +y)(2x +5)-(x +y)(x +3) = (x +y){(2x +5)-(x +3)} = (x +y)(2x +5 -x -3) = (x +y)(x +2)
Q u e s t i o n : 1 7
Factorise:
ar + br + at + bt
S o l u t i o n :
By grouping the terms:
ar+br+at +bt = (ar+br)+(at +bt) = r(a +b)+t(a +b) = (a +b)(r+t)
? ar+br+at +bt = (a +b)(r+t)
Q u e s t i o n : 1 8
Factorise:
x
2
- ax - bx + ab
S o l u t i o n :
By suitably arranging the terms:
x
2
-ax -bx +ab = x
2
-bx -ax +ab = x
2
-bx -(ax -ab) = x(x -b)-a(x -b) = (x -b)(x -a)
? x
2
-ax -bx +ab = (x -b)(x -a)
Q u e s t i o n : 1 9
Factorise:
ab
2
- bc
2
- ab + c
2
S o l u t i o n :
By suitably arranging the terms:
ab
2
-bc
2
-ab +c
2
= ab
2
-ab -bc
2
+c
2
= ab
2
-ab - bc
2
-c
2
= ab(b -1)-c
2
(b -1) = (b -1) ab -c
2
? ab
2
-bc
2
-ab +c
2
= (b -1) ab -c
2
Q u e s t i o n : 2 0
Factorise:
x
2
- xz + xy - yz
S o l u t i o n :
By suitably arranging the terms:
x
2
-xz +xy -yz = x
2
+xy -xz -yz = x
2
+xy -(xz +yz) = x(x +y)-z(x +y) = (x +y)(x -z)
? x
2
-xz +xy -yz = (x +y)(x -z)
Q u e s t i o n : 2 1
Factorise:
6ab - b
2
+ 12ac - 2bc
S o l u t i o n :
By suitably arranging the terms:
6ab -b
2
+12ac -2bc = 6ab +12ac -b
2
-2bc = (6ab +12ac)- b
2
+2bc = 6a(b +2c)-b(b +2c) = (b +2c)(6a -b)
? 6ab -b
2
+12ac -2bc = (b +2c)(6a -b)
Q u e s t i o n : 2 2
Factorise:
(x - 2y)
2
+ 4x - 8y
S o l u t i o n :
We have:
(x -2y)
2
+4x -8y = (x -2y)
2
+4(x -2y) = (x -2y)(x -2y)+4(x -2y) = (x -2y){(x -2y)+4} = (x -2y)(x -2y +4)
? (x -2y)
2
+4x -8y = (x -2y)(x -2y +4)
Q u e s t i o n : 2 3
Factorise:
y
2
- xy(1 - x) - x
3
S o l u t i o n :
We have:
y
2
-xy(1 -x)-x
3
= y
2
-xy +x
2
y -x
3
= y
2
-xy + x
2
y -x
3
= y(y -x)+x
2
(y -x) = (y -x) y +x
2
? y
2
-xy(1 -x)-x
3
= (y -x) y +x
2
Q u e s t i o n : 2 4
Factorise:
( )
( ) ( ) ( )
( )
( )
( )
( ) ( ) ( )
( )
(ax + by)
2
+ (bx - ay)
2
S o l u t i o n :
We have:
(ax +by)
2
+(bx -ay)
2
= a
2
x
2
+b
2
y
2
+2axby + b
2
x
2
+a
2
y
2
-2bxay = a
2
x
2
+a
2
y
2
+b
2
y
2
+b
2
x
2
+2axby -2bxay = a
2
x
2
+y
2
+b
2
x
2
+b
2
y
2
+2axby -2axb
? (ax +by)
2
+(bx -ay)
2
= x
2
+y
2
a
2
+b
2
Q u e s t i o n : 2 5
Factorise:
ab
2
+ (a - 1)b - 1
S o l u t i o n :
We have:
ab
2
+(a -1)b -1 = ab
2
+ba -b -1 = ab
2
+ba -(b +1) = ab(b +1)-1(b +1) = (b +1)(ab -1)
? ab
2
+(a -1)b -1 = (b +1)(ab -1)
Q u e s t i o n : 2 6
Factorise:
x
3
- 3x
2
+ x - 3
S o l u t i o n :
We have:
x
3
-3x
2
+x -3 = x
3
-3x
2
+(x -3) = x
2
(x -3)+1(x -3) = (x -3) x
2
+1
? x
3
-3x
2
+x -3 = (x -3) x
2
+1
Q u e s t i o n : 2 7
Factorise:
ab(x
2
+ y
2
) - xy(a
2
+ b
2
)
S o l u t i o n :
We have:
ab x
2
+y
2
-xy a
2
+b
2
= abx
2
+aby
2
-a
2
xy -b
2
xy = abx
2
-a
2
xy +aby
2
-b
2
xy = ax(bx -ay)+by(ay -bx) = ax(bx -ay)-by(bx -ay)
? ab x
2
+y
2
-xy a
2
+b
2
= (bx -ay)(ax -by)
Q u e s t i o n : 2 8
Factorise:
x
2
- x(a + 2b) + 2ab
S o l u t i o n :
We have:
x
2
-x(a +2b)+2ab = x
2
-ax -2bx +2ab
= x
2
-2bx -ax +2ab = x
2
-2bx -(ax -2ab) = x(x -2b)-a(x -2b) = (x -2b)(x -a)
? x
2
-x(a +2b)+2ab = (x -2b)(x -a)
( ) ( ) ( )
( ) ( )
( )
( ) ( )
( )
( ) ( )
( ) ( )
( )
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FAQs on RS Aggarwal Solutions: Exercise 7A - Factorisation - Mathematics (Maths) Class 8
| 1. What is factorisation and why is it important in mathematics? |  |
Ans. Factorisation is the process of expressing a mathematical expression as a product of its factors. It is important in mathematics as it helps in simplifying complex expressions, finding common factors, and solving equations more easily.
| 2. How do you factorise a quadratic expression? | |
Ans. To factorise a quadratic expression, we look for two numbers that multiply to give the constant term and add up to give the coefficient of the middle term. We then use these numbers to split the middle term and factorise the expression.
| 3. Can all quadratic expressions be factorised? | |
Ans. No, not all quadratic expressions can be factorised. Some quadratic expressions are prime and cannot be expressed as a product of linear factors. In such cases, we can use other methods like completing the square or using the quadratic formula to solve the expression.
| 4. What are the different methods of factorisation? | |
Ans. There are several methods of factorisation, including:
- Common Factor Method: Factoring out the common factor from all terms of the expression.
- Difference of Squares Method: Factoring expressions in the form of a^2 - b^2 as (a + b)(a - b).
- Perfect Square Trinomial Method: Factoring expressions in the form of a^2 + 2ab + b^2 or a^2 - 2ab + b^2 as (a + b)^2 or (a - b)^2.
- Grouping Method: Grouping terms in pairs and finding the common factors in each pair.
- Trial and Error Method: Trying different combinations of factors to find the correct factorisation.
| 5. How can factorisation be applied in real-life situations? | |
Ans. Factorisation can be applied in various real-life situations, such as:
- Simplifying complex financial equations: Factorisation can help simplify equations related to interest rates, investments, or loans, making them easier to understand and solve.
- Solving engineering problems: Factorisation can be used to simplify equations in engineering calculations, such as determining the optimal size of components or analyzing circuit networks.
- Factoring large numbers: Factorisation plays a crucial role in cryptography and computer security by helping to find prime factors of large numbers, which is essential for encryption algorithms.
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