Question

Factorise the following:

1) x^2-4x-4y-y^2

2)25x^2-10x+1-36y^2

3)x^2+2xy+y^2-a^2+2ab-b^2

4) 5x^2-16x-21

Answer 1

(1)x2 - 4x + 4y - y2 = x2  - y2  - 4x +  4y
​                           = (x + y)(x - y) - 4(x - y)
​                           = (x - y)(x+ y - 4)

(2)(25x2 - 10x + 1) - 36y2

= (5x - 1)2 - 36y2 (by the formula.. a2- b2= a2 + b2- 2ab)

= (5x-1)2 - (6y)2

= (5x - 1 + 6y) (5x - 1 - 6y)

(3)x2 - 2xy + y2 - a2 - 2ab - b2

On regrouping,

(x2 + y2 - 2xy) - (a2 + b2 + 2ab)

(x-y)2 - (a+b)2

We know that a2 - b2 = (a+b)(a-b)

(x-y+a+b)(x-y-a-b)    ans.

(4)first you multiply -21 by coefficient of x^2 i.e 5
So you have -21 x 5
These have to be factored in such a way their sum becomes as - 16 i.e coefficient of x
The apt factors are -21 and 5
Now divide these by 5 and you have - 21/5 and 5/5 ie. 1/1
Now multiply the denominator by x and affix the numerator with that
So 5x-21 is one factor and x + 1 is the other factor
So (5x-21)(x+1) is the expected one