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

Factoring There are various 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.

What is a 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: Finding a GCF Example 1: Find the GCF of each set of monomials. 8, 12 b) 10, 21 c) 24, 60, 36 4 1 12

Ex 1: Finding GCFs 2x2 6a2b 3xy2 4x 3x2

Factoring a Polynomial w/GCF Determine what the GCF of the terms is, and factor that out Rewrite the expression using the distributive property

Ex 2: Factoring By Distributive Property Factor each polynomial.

Try these.

Grouping Grouping is a factoring technique used when a polynomial contains four or more terms.

Steps for Factoring By Grouping Group terms with common factors (separate the polynomial expression into the sum of two separate expressions) Factor the GCF out of each expression Rewrite the expression using the distributive property (factor into a binomial multiplied by a binomial)

Example 3: Factor each polynomial.

Ex 3: Factor each polynomial.

Ex 3: Continued.

Ex 3: Cont.

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.]

Steps to factor a Trinomial Steps for factoring a trinomial Multiply a * 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 two factors. 4) Factor by grouping.

Ex4: Factoring Trinomials

Example 4: Factor each polynomial

Try some more…

Try some more…

Try these.

More Examples

More Examples

Look For GCF first! 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.

Ex 5: Factor each polynomial

Ex 5: Factor each polynomial

Ex 5: GCF first!

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.

Factoring Differences of Squares

Factoring Differences of Squares

Factoring Differences of Squares

Factoring Differences of Squares GCF first!!

Factoring Differences of Squares

Factoring Differences of Squares

Sum/Difference of Cubes

Sum/Difference of Cubes

Try these

Try these

Try these

Simplifying Polynomial Quotients In the previous section (5-3), we learned to simplify the quotient of two polynomials using long division or synthetic division. 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.

Ex1: Simplify Factor Numerator and Denominator! Eliminate Common Factors in Numerator and Denominator!

Ex 2: Simplify

Ex 3: Simplify In order to eliminate common factors , one must be in the numerator an the other in the denominator. This expression cannot be simplified further…

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