Tuesday, August 8, 2017

98-GRE Math Test - Important Key points and formulas Part - 3

Blog-98

In continuation of Blog-97, 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: 

Here following 4 possibilities can be studied to understand the factorization in better way.

1) (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
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

3) (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

Note: Here formulae 2 and 3 are of same type. the coefficient of the middle term is the difference of the Constance and the sign is to be taken of the greater Constance. 

4) (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

Now we will study these types in detail:

1) (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 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).
In the next part we will see remaining 3 types in detail. These 3 types are given below.

2) (x + a) (x - b) = x 2 + (a - b) x - ab
3) (x - a) (x + b) = x 2 - (a - b) x - ab
4) (x - a) (x - b) = x 2 - (a + b) x + ab