# Factoring Polynomials

The process of factoring a polynomial is analogous to the
process of factoring

an integer into its prime factors. For example 180 factors as 2^{2}3^{2}5. We want

to do the same thing, but with polynomials.

** Factoring out the Greatest Common Factor
**Example.

**Factoring by Grouping**

Method : Group terms in pairs depending on whether they have common

factors, then factor out common term .

Example.

Example. −10z + 6yz + 5 − 3y

**Factoring polynomials of the form**ax

^{2}+ bx + c

Polynomials of this form (degree 2) are called

**quadratic polynomials .**

Method: Find two numbers whose sum is b and whose product is ac. Replace

b by the sum of these two numbers, and factor the resulting polynomial by

grouping.

Example. 12b

^{2}+ 17b + 6

Example. x

^{2 }− 4x − 12

**Factoring Special Products **

1) Difference of Two Squares a ^{2} − b^{2} = (a + b)(a − b)

2) Perfect Square Trinomial a^{2} + 2ab + b^{2} = (a + b)^{2
}

3) Perfect Square Trinomial a ^{2} − 2ab + b^{2} = (a − b)^{2
}

4) Difference of Two Cubes a ^{3}−b^{3} = (a−b)(a^{2}+ab+b^{2})

5) Sum of Two Cubes a ^{3}+b^{3} = (a+b)(a^{2}−ab+b^{2})

Example. 9t^{2} − v^{2
}

Example. 4w^{2} − 4w + 1

Example. 9x^{2} − 12xy + 4y^{2
}

Example. m^{3} + n^{3
}

Example. a^{3} − 8

Example. 8t^{3}h^{3} + n^{9}

**Factoring Completely
**

Example. a

^{4}b

^{2}− 16b

^{2}

Example. a

^{3}+ a

^{2}− 4a − 4

Example. −36x

^{3}+ 18x

^{2}+ 4x

Example. a

^{7}− a

^{6}− 64a + 64

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