Section P5 Factoring Polynomials

Common Factors

Factoring a polynomial containing the sum of monomials mean finding an equivalent expression that is a product. In this section we will be factoring over the set of integers, meaning that the coefficients in the factors are integers. Polynomials that cannot be factored using integer coefficients are called prime.

Example Factor:

Factoring by Grouping

Sometimes all of the terms of a polynomial may not contain a common factor. However, by a suitable grouping of terms it may be possible to factor. This is called factoring by grouping.

Example Factor by Grouping:

Example Factor by Grouping:

Factoring Trinomials

+ + Factors of 8 8,1 4,2 -8,-1 -4,-2 Sum of Factors 9 6 -9 -6 4 2 Choose either two positive or two negative factors since the sign in front of the 8 is positive.

- + 1 5 Factor: Possible factorizations Sum of outside and inside products 1 5 Since the sign in front of the 5 is a negative, one factor will be positive and one will be negative.

Possible Factorizations Sum of Inside and Outside products Example Factor: Possible Factorizations Sum of Inside and Outside products

Example Factor: Possible Factorizations Sum of Inside and outside Products

Factoring the Difference of Two Squares

Repeated Factorization- Another example Can the sum of two squares be factored?

Example Factor Completely:

Example Factor Completely:

Example Factor Completely:

Factoring Perfect Square Trinomials

Example Factor:

Example Factor:

Factoring the Sum and Difference of Two Cubes

Example Factor:

Example Factor:

Example Factor:

A Strategy for Factoring Polynomials

Example Factor Completely:

Example Factor Completely:

Example Factor Completely:

Example Factor Completely:

Example Factor Completely:

Example Factor Completely:

Factoring Algebraic Expressions Containing Fractional and Negative Exponents

Expressions with fractional and negative exponents are not polynomials, but they can be factored using similar techniques. Find the greatest common factor with the smallest exponent in the terms.

Example Factor and simplify:

Example Factor and simplify:

Example Factor and simplify:

Factor Completely: (a) (b) (c) (d)

Factor Completely: (a) (b) (c) (d)