## Blog-99

In continuation of**Blog-98**, we will see all the important formulas and useful statements which are to be used in Math test of GRE.

### Factorization (Continued):

#### b) Factorization of Polynomial (Continued):

2) (x + a) (x - b) = x (x - b) + a (x - b)= x

^{2}- b x + a x - ab

= x

^{2}+ a x - b x - ab

= x

^{2}+ (a - b) x - ab

Generally we call

**x**

^{2 }as the first term,

**(a + b) x**as the middle term and

**ab**as the last term.

### Basic concept:

a) Step-1: See the sign of the last term.b) Step-2: Here it is "-" so factorize the product of the coefficient of first term (here it is 1) and the last term in such a way that the

**DIFFERENCE**of these two factors must be the coefficient of the middle term.

c) Step-3: Get the factors.

### Example-1:

Factorize: x^{2}+ 4 x - 21.

a) Step-1: Here sign of the last term 21 is "-"

b) Step-2: The coefficient of the first term is 1 and the last term is 21 , so the product of 1 and 21 is 21. Now the factors of 21 are 3 and 7 and as the sign of the last term is "-", their subtraction is 7 - 3 = 4 which is the coefficient of the middle term. (Note: Here, the coefficient of the middle term is positive so we took it as 7 - 3).

= x

^{2}+ 4 x - 21

= x

^{2}+ (7 - 3) x - (3 x 7)

=

__x__

__-__

^{2}+ 7 x__3 x__

__- (3 x 7)__

=

__x__

__(x + 7)__-

__3 (x__

__+ 7)__

= (x - 3) (x + 7)

c) Step-3: So the factors of x

^{2}+ 4 x - 21 are (x - 3) and (x + 7)

### Example-2:

Factorize: 8 x^{2}+ 18 x - 5.

a) Step-1: Here sign of the last term 5 is "-"

b) Step-2: The coefficient of the first term is 8 and the last term is 5 , so the product of 8 and 5 is 8 X 5. Now the factors of 8 X 5 are 2, 2, 2 and 5 and as the sign of the last term is "-", so, we take two factor in such a way that their difference will be 18. Here 2, 2, 2 and 5 will give us 2 and 20. So, here subtraction is 20 - 2 = 18 which is the coefficient of the middle term.

= 8 x

^{2}+ 18 x - 5

= 8 x

^{2}+ (20 - 2) x - 5

=

__8 x__

__-__

^{2}+ 20 x__2 x__

__- 5__

=

__4 x__

__(2 x + 5)__-

__1 (2 x__

__+ 5)__

= (2 x + 5) (4 x - 1)

c) Step-3: So the factors of 8 x

^{2}+ 18 x + 5 are (2 x + 5) and (4 x - 1).

In the next part we will see remaining 2 types in detail. These 2 types are given below.

3) (x - a) (x + b) = x

4) (x - a) (x - b) = x

3) (x - a) (x + b) = x

^{2}- (a - b) x - ab4) (x - a) (x - b) = x

^{2}- (a + b) x + ab