In continuation of part - 5, 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):
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.
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 SUM of these two factors must be the coefficient of the middle term.
c) Step-3: Get the factors.
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)
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 factors 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).
= 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 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 factors 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 a few examples and some essential formulae.
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