5-4 Factoring Polynomials Objectives: Students will be able to: Factor polynomials Simplify polynomial quotients by factoring

Factoring There are several different techniques used to factor polynomials. The technique(s) used depend on the number of terms in the polynomial, and what those terms are. Throughout this section we will examine different factoring techniques and how to utilize one or more of those techniques to factor a polynomial.

GCF Greatest common factor (gcf): largest factor that all terms have in common You can find the GCF for a polynomial of two or more terms.

Example 1: Factor each polynomial. To factor a polynomial expression using GCF: 1) determine what the GCF of the terms is, and factor that out 2) rewrite the expression using the distributive property Example 1: Factor each polynomial.

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Grouping Grouping is a factoring technique used when a polynomial contains four or more terms. Steps for factoring by grouping (based on a polynomial of four terms) 1) group terms with common factors (separate the polynomial expression into two separate expressions) 2) factor the GCF out of each expression 3) rewrite the expression using the distributive property (factor into a binomial multiplied by a binomial)

Example 2: Factor each polynomial.

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Factoring Trinomials The standard form for a trinomial is: The goal of factoring a trinomial is to factor it into two binomials. [If we re-multiplied the binomials together, that should get us back to the original trinomial.] In order to factor, we need to “bust” our b term.

Steps for factoring a trinomial: Multiply a X c 2) Look for factors of the product in step 1 that add to give you the ‘b’ term. 3) Rewrite the ‘b’ term using these factors. 4) Factor by grouping.

Example 3: Factor each polynomial

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There are instances when a polynomial will have a GCF that can be factored out first. Doing so will make factoring a trinomial much easier. Example 4: Factor each polynomial

Now that you are factoring experts, try this one.

Additional Factoring Techniques There are certain binomials that are factorable, but cannot be factored using any of the previous factoring techniques. These binomials deal with perfect square factors or perfect cube factors.

Example 5: Factor each polynomial. Try these.

Some quotients can be simplified using factoring. To do so: 1) factor the numerator (if possible) 2) factor the denominator (if possible) 3) reduce the fraction Tip**: Be sure to check for values that the variable cannot equal. Remember that the denominator of a fraction can never be zero.

Example 6: Simplify.

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To recap: Always try and factor out a GCF first, if possible. It will make life much easier.