To find factors of polynomial, we use the following ways.
(i) Using grouping method
(ii) Using algebraic identities
(iii) Using synthetic division
Some times we may use more than one of the methods given above.
Taking the common values out, this method is known as grouping method.
The given question will be in the form of
The expansion will be in any one of the form, for example
= x^{2} + 10x + 25
= x^{2} + 2 (x) (5) + 5^{2}
= (x + 5)^{2}
So, the factors are (x + 5)(x + 5).
The polynomial which is having the highest exponent of 3 or more than 3, we can use synthetic division and find factors.
To see more example problems,
Factor the polynomial into linear factors.
Problem 1 :
x³ - 4x
Solution :
= x³ - 4x
Using grouping method, factoring x we get
= x(x² - 4)
x² - 4 can be written as
= x(x + 2) (x - 2)
Problem 2:
6x² - 54
Solution :
= 6x² - 54
= 6(x² - 9)
x² - 9 can be written as
= 6(x + 3) (x - 3)
Problem 3 :
4x² + 8x - 60
Solution :
= 4x² + 8x - 60
= 4(x² + 2x - 15)
= 4(x - 3) (x + 5)
Problem 4 :
15x³ - 22x² + 8x
Solution :
= 15x³ - 22x² + 8x
= x (15x² - 22x + 8)
= x (15x² - 12x - 10x + 8)
= x [3x(5x - 4) - 2(5x - 4)]
= x(3x - 2) (5x - 4)
Problem 5:
x³ + 2x² - x - 2
Solution :
= x³ + 2x² - x - 2
By grouping,
= (x³ + 2x²) + (-x - 2)
= x²(x + 2) - 1(x + 2)
= (x + 2) (x² - 1)
x² - 1 can be written as
= (x - 1) (x + 1) (x + 2)
Problem 6 :
x^{4 }+ x³ - 9x² - 9x
Solution :
= x^{4 }+ x³ - 9x² - 9x
= x(x³ + x² - 9x - 9)
By grouping,
= x(x³ - 9x) + (x² - 9)
= x [x(x² - 9) + 1(x² - 9)]
= x [(x + 1) (x² - 9)]
x² - 9 can be written as
= x(x + 1) (x - 3) (x + 3)
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