In continuation of part - 3, we will see all the important formulas and useful statements which are to be used in the Math test GRE.

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Factorization (Continued):

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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.

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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 the 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.

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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)

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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 the two factors 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 the remaining 2 types in detail. These 2 types are given below.

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

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

In the next part, we will see a few examples and some essential formulae.