Factorisation
Class 8
What are Factorisation
When a number is written as product of its factore we call it the factorisation of the number.
For finding factors we have three methods
1. Finding out common factors
2. Factorising Binomials Completely
3. Factorising Trinomials Completely
4. Dividing after factorising trinomials
Factorisation by finding common factors
1. 21a2b + 42ab2
Solution :-
21a2b = 3 x 7 x a x a x b
42ab2 = 7 x 7 x a x b x b
Therefore, 21a2b + 42ab2
= (3 x 7 x a x a x b) + (7 x 7 x a x b x b)
= (7ab x 3a) + (7ab x 7b)
= 7ab (3a + 7b)
2. 4x2y2 + 10x2y - 8xy2
Solution :- The common factors are 2 and xy
so 4x2y2 = 2 x 2 x x x x x y x y
10x2y = 2 x 5 x x x x x y
8xy2 = 2 x 2 x 2 x x x y x y
Therefore 2xy(2xy + 5x - 4y)
Ans :- 2xy(2xy + 5x - 4y)
Factorising Binomials Completely
Eg.Factorise completely
1. x4 – 16
In this question, we will have to make it into an identity.
(x2)2 – (4)2
{(x2 + 4)}{(x2 – 4)}
Now, we can see that it can again be made into an identity but this time, notice that we are converting the second part to an identity because the first cannot be converted into an identity.
{x2 + 4)}{(x)2 – (2)2}
Ans :- {x2 + 4}{x+2}{x – 2}
2. 3y4 – 108y2
In this question, will have to factorise it first and then make into an identity.
Here, we can take 3y2 as the common factor and keep the rest all in a bracket.
3y2 (y2 – 36)
3y2 {(y)2 – (6)2}
Ans :- 3y2 {(y + 6)(y – 6)}
Factorising Trinomials Completely
Eg. Factorise completely
1. x2 + 11x + 24
In how many ways can we write 24
2 X 12, 3 X 8, 4 X 6, 24 X 1 now take two number those sum is 11 and when we multiply this number we get 24
x2 + 8x + 3x +24
x (x + 8) + 3 (x + 8)
Ans :- (x + 3)(x + 8)
2. 6a2 – 13ab + 2b2
6 X 2 = 12
In how many ways can we write 12
2 X 6, 3 X 4, 12 X 1 now take two number those sum is 13 and when we multiply this number we get 12
6a2 – 12ab – ab + 2b2
6a (a – 2b) – b (a – 2b)
Ans :- (6a – b)(a – 2b)
Dividing after factorising trinomials
1. 9a2 + 12a + 4 by (3a + 2)
9 X 4 = 36
In how many ways can you write 36
2 X 18, 3 X 12, 4 X 9, 6 X 6, 36 X 1 now take two number those sum is 12 and when we multiply this number we get 36
9a2 + 6a + 6a + 4
3a (3a + 2) + 2 (3a + 2)
Now, Divide (3a + 2)(3a + 2) by (3a + 2)
(3a + 2)(3a + 2)
(3a + 2)
Ans :- 3a + 2
2. 2x2 – 7x + 5 by (x – 1)
Now, 2 X 5 = 10
In how many ways can you write 10
2 X 5, 10 X 1 now take two number those sum is 7 and when we multiply we get 10
2x2 – 5x – 2x + 5
x (2x - 5) – 1 (2x – 5)
Now, Divide (x – 1)(2x – 5) by (x - 1)
(x – 1)(2x – 5)
(x – 1)
Ans :- 2x - 5
When a number is written as product of its factore we call it the factorisation of the number.
For finding factors we have three methods
1. Finding out common factors
2. Factorising Binomials Completely
3. Factorising Trinomials Completely
4. Dividing after factorising trinomials
Factorisation by finding common factors
1. 21a2b + 42ab2
Solution :-
21a2b = 3 x 7 x a x a x b
42ab2 = 7 x 7 x a x b x b
Therefore, 21a2b + 42ab2
= (3 x 7 x a x a x b) + (7 x 7 x a x b x b)
= (7ab x 3a) + (7ab x 7b)
= 7ab (3a + 7b)
2. 4x2y2 + 10x2y - 8xy2
Solution :- The common factors are 2 and xy
so 4x2y2 = 2 x 2 x x x x x y x y
10x2y = 2 x 5 x x x x x y
8xy2 = 2 x 2 x 2 x x x y x y
Therefore 2xy(2xy + 5x - 4y)
Ans :- 2xy(2xy + 5x - 4y)
Factorising Binomials Completely
Eg.Factorise completely
1. x4 – 16
In this question, we will have to make it into an identity.
(x2)2 – (4)2
{(x2 + 4)}{(x2 – 4)}
Now, we can see that it can again be made into an identity but this time, notice that we are converting the second part to an identity because the first cannot be converted into an identity.
{x2 + 4)}{(x)2 – (2)2}
Ans :- {x2 + 4}{x+2}{x – 2}
2. 3y4 – 108y2
In this question, will have to factorise it first and then make into an identity.
Here, we can take 3y2 as the common factor and keep the rest all in a bracket.
3y2 (y2 – 36)
3y2 {(y)2 – (6)2}
Ans :- 3y2 {(y + 6)(y – 6)}
Factorising Trinomials Completely
Eg. Factorise completely
1. x2 + 11x + 24
In how many ways can we write 24
2 X 12, 3 X 8, 4 X 6, 24 X 1 now take two number those sum is 11 and when we multiply this number we get 24
x2 + 8x + 3x +24
x (x + 8) + 3 (x + 8)
Ans :- (x + 3)(x + 8)
2. 6a2 – 13ab + 2b2
6 X 2 = 12
In how many ways can we write 12
2 X 6, 3 X 4, 12 X 1 now take two number those sum is 13 and when we multiply this number we get 12
6a2 – 12ab – ab + 2b2
6a (a – 2b) – b (a – 2b)
Ans :- (6a – b)(a – 2b)
Dividing after factorising trinomials
1. 9a2 + 12a + 4 by (3a + 2)
9 X 4 = 36
In how many ways can you write 36
2 X 18, 3 X 12, 4 X 9, 6 X 6, 36 X 1 now take two number those sum is 12 and when we multiply this number we get 36
9a2 + 6a + 6a + 4
3a (3a + 2) + 2 (3a + 2)
Now, Divide (3a + 2)(3a + 2) by (3a + 2)
(3a + 2)(3a + 2)
(3a + 2)
Ans :- 3a + 2
2. 2x2 – 7x + 5 by (x – 1)
Now, 2 X 5 = 10
In how many ways can you write 10
2 X 5, 10 X 1 now take two number those sum is 7 and when we multiply we get 10
2x2 – 5x – 2x + 5
x (2x - 5) – 1 (2x – 5)
Now, Divide (x – 1)(2x – 5) by (x - 1)
(x – 1)(2x – 5)
(x – 1)
Ans :- 2x - 5