100-GRE Math-5-Important Key points and formulas

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In continuation of part - 4, we will see all the important formulas and useful statements which are to be used in the Math test GRE.

Factorization (Continued):

b) Factorization of Polynomial (Continued): 

3) (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 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.

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 3 - 7 = -4 which is the coefficient of the middle term. (Note: Here, the coefficient of the middle term is positive so we took it as 3 - 7).  
                         = x ² - 4 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 - 7) (x + 3)
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 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 2 - 20 = -18 which is the coefficient of the middle term.  
                         = 8 x ² - 18 x - 5
                         = 8 x ² + (2 - 20) x - 5
                         = 8 x ² + 2 x - 20 x - 5
                         = 2 x (4 x + 1) - 5 (4 x + 1)
                         = (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 1 type in detail. These 1 types are given below.
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.

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ANIL SATPUTE