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Chapter 3 3-4 factoring polynomials

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

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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.

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Factor theorem

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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 4 + 6 x 3 – 5 x – 10)

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

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

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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.

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Factoring by Grouping Factor: x 3 – x 2 – 25 x + 25. Factor: 2 x 3 + x 2 + 8 x + 4.

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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.

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Factoring the Sum or Difference of Two Cubes Factor the expression. 4 x 4 + 108 x 125 d 3 – 8

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

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Geometry Application The volume of a plastic storage box is modeled by the function V ( x ) = x 3 + 6 x 2 + 3 x – 10. Identify the values of x for which V ( x ) = 0, then use the graph to factor V ( x ).

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Geometry application The volume of a rectangular prism is modeled by the function V ( x ) = x 3 – 8 x 2 + 19 x – 12, which is graphed below. Identify the values of x for which V ( x ) = 0, then use the graph to factor V ( x ).

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Homework Do odd problems from 17-32 in your book page 177

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

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