§ 5.6 A General Factoring Strategy

Blitzer, Intermediate Algebra, 5e – Slide #2 Section 5.6 Factoring a Polynomial We have looked at factoring out a common factor, factoring by grouping, factoring a difference of squares, factoring general trinomials using trial and error, factoring a sum or difference of cubes, and factoring other special forms. It is important when you wish to factor a polynomial to know where to start. You should always look first to see if there is a common factor that you can factor out. Do that first. Then consider the number of terms in the polynomial. Strategies for factoring a polynomial based on the number of terms in the polynomial follow.

Blitzer, Intermediate Algebra, 5e – Slide #3 Section 5.6 A Strategy for Factoring Polynomials A Strategy for Factoring a Polynomial 1) If there is a common factor, factor out the GCF or factor out a common factor with a negative coefficient. 2) Determine the number of terms in the polynomial and try factoring as follows: (a) If there are two terms, can the binomial be factored by using one of the following special forms. Difference of two squares: Sum of two cubes: Difference of two cubes:

Blitzer, Intermediate Algebra, 5e – Slide #4 Section 5.6 A Strategy for Factoring Polynomials A Strategy for Factoring a Polynomial (b) If there are three terms, is the trinomial a perfect square trinomial? If so, factor by using one of the following forms: If the trinomial is not a perfect square trinomial, try factoring by trial and error or grouping. (c) If there are four or more terms, try factoring by grouping. 3) Check to see if any factors with more than one term in the factored polynomial can be factored further. If so, factor completely. Remember to check the factored form by multiplying or by using the TABLE or GRAPH feature of a graphing utility. CONTINUED

Blitzer, Intermediate Algebra, 5e – Slide #5 Section 5.6 Factoring a PolynomialEXAMPLE SOLUTION Factor: 1) If there is a common factor, factor out the GCF. Because 3 is common to both terms, we factor it out. Factor out the GCF 2) Determine the number of terms and factor accordingly. The factor has two terms. This binomial can be expressed as, so it can be factored as the difference of two cubes.

Blitzer, Intermediate Algebra, 5e – Slide #6 Section 5.6 Factoring a Polynomial 3) Check to see if factors can be factored further. In this case, they cannot, so we have factored completely. Rewrite as the difference of two cubes Simplify CONTINUED Factor

Blitzer, Intermediate Algebra, 5e – Slide #7 Section 5.6 Factoring a PolynomialEXAMPLE SOLUTION Factor: 1) If there is a common factor, factor out the GCF. Because 2y is common to both terms, we factor it out. Factor out the GCF 2) Determine the number of terms and factor accordingly. The factor has two terms. This binomial can be expressed as, so it can be factored as the difference of two squares.

Blitzer, Intermediate Algebra, 5e – Slide #8 Section 5.6 Factoring a Polynomial 3) Check to see if factors can be factored further. We note that is the difference of two squares,, so we continue factoring. Rewrite as the difference of two squares CONTINUED Factor

Blitzer, Intermediate Algebra, 5e – Slide #9 Section 5.6 Factoring a Polynomial The previous factorization CONTINUED Rewrite last factor as the difference of two squares Factor

Blitzer, Intermediate Algebra, 5e – Slide #10 Section 5.6 Factoring a PolynomialEXAMPLE SOLUTION Factor: 1) If there is a common factor, factor out the GCF. Because 5 is common to both terms, we factor it out. Factor out the GCF 2) Determine the number of terms and factor accordingly. The factor has three terms and is a perfect square trinomial. We factor using.

Blitzer, Intermediate Algebra, 5e – Slide #11 Section 5.6 Factoring a Polynomial Factor out the GCF 3) Check to see if factors can be factored further. In this case, they cannot, so we have factored completely. CONTINUED Rewrite the part in parentheses in Factor form