## Blog-101

In continuation of**Blog-100**, 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):

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

^{2}- b x - a 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

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

c) Step-3: Get the factors.

### Example-1:

Factorize: x^{2}- 10 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 addition is -3 - 7 = -10 which is the coefficient of the middle term.

= x

^{2}- 10 x + 21

= x

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

=

__x__

__-__

^{2}- 3 x__7 x__

__+ (3 x 7)__

=

__x__

__(x - 3)__-

__7 (x__

__- 3)__

= (x - 3) (x - 7)

c) Step-3: So the factors of x

^{2}- 10 x + 21 are (x - 3) and (x - 7)

### Example-2:

Factorize: 8 x^{2}- 14 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 sum will be 14. Here 2, 2, 2 and 5 will give us 4 and 10. So, here addition is -4 - 10 = -14 which is the coefficient of the middle term.

= 8 x

^{2}- 14 x + 5

= 8 x

^{2}- (4 + 10) x + 5

=

__8 x__

__-__

^{2}- 4 x__10 x__

__+ 5__

=

__4 x__

__(2 x - 1)__-

__5 (2 x__

__- 1)__

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

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

^{2}- 14 x + 5 are (2 x - 1) and (4 x - 5).