Chapter factoring polynomials

Objectives Use the Factor Theorem to determine factors of a polynomial. Factor the sum and difference of two cubes.

Factor Theorem  Recall that if a number is divided by any of its factors, the remainder is 0. Likewise, if a polynomial is divided by any of its factors, the remainder is 0.  The Remainder Theorem states that if a polynomial is divided by (x – a), the remainder is the value of the function at a. So, if (x – a) is a factor of P(x), then P(a) = 0.

Factor theorem

Example#1  Determine whether the given binomial is a factor of the polynomial P ( x ).  A. ( x + 1); ( x 2 – 3 x + 1)  B. ( x + 2); (3 x x 3 – 5 x – 10)

Example#2  Determine whether the given binomial is a factor of the polynomial P ( x ).  a. ( x + 2); (4 x 2 – 2 x + 5)

Student guided practice  Do problems 1-3 in your book page 177

Factoring  You are already familiar with methods for factoring quadratic expressions. You can factor polynomials of higher degrees using many of the same methods you learned.

Factoring by Grouping  Factor: x 3 – x 2 – 25 x  Factor: 2 x 3 + x x + 4.

Factoring polynomials  Just as there is a special rule for factoring the difference of two squares, there are special rules for factoring the sum or difference of two cubes.

Factoring the Sum or Difference of Two Cubes  Factor the expression.  4 x x  125 d 3 – 8

Student guided practice  Do problems 4-8 in your book page 176

Geometry Application  The volume of a plastic storage box is modeled by the function V ( x ) = x x x – 10. Identify the values of x for which V ( x ) = 0, then use the graph to factor V ( x ).

Geometry application  The volume of a rectangular prism is modeled by the function V ( x ) = x 3 – 8 x x – 12, which is graphed below. Identify the values of x for which V ( x ) = 0, then use the graph to factor V ( x ).

Homework  Do odd problems from in your book page 177

closure  Today we learned about polynomials  Next class we are going to learned about finding real roots in polynomials