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Warm Up Factor each expression.

Learn how to find factors of polynomials, factor sums and differences of cubes, and use Factor Theorem and Remainder Theorem effectively in this comprehensive lesson.

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Warm Up Factor each expression.

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  1. Warm Up Factor each expression. 1. 3x – 6y 2. a2 – b2 Find each product. 4. (a + 1)(a2 + 1) 3. (x – 1)(x + 3) LEARNING GOALS – LESSON 6.4 6.4.1: Use the Factor Theorem to determine factors of a polynomial. 6.4.2: Factor the sum and difference of two cubes. Recall: If a number is divided by any of its factors, the remainder is ____. Likewise, if a polynomial is divided by any of its factors, the remainder is _____. The Remainder Theorem states: If a polynomial is divided by (x – a), the remainder is the value of the function at a. RECALL: If (x – a) is a factor of P(x), then P(___)=___ Example 1: Determining Whether a Linear Binomial is a Factor Determine whether the given binomial is a factor of P(x). A. (x + 1); (x2 – 3x + 1) B. (x + 2); (3x4 + 6x3 – 5x – 10) Find P(___) by synthetic substitution. Find P(____) by synthetic substitution. P(____) = ____ P(____) = ____

  2. Example 1C: Determining Whether a Linear Binomial is a Factor Determine whether the given binomial is a factor of P(x). C. (3x – 6); (3x4 – 6x3 + 6x2 + 3x – 30) Divide the polynomial by 3, then find P(2) by synthetic substitution. P(____) = _____, so (3x – 6) ______________________________________________________P(x) = 3x4 – 6x3 + 6x2 + 3x – 30. You can also FACTOR a polynomial to find its factors. Example 2A: Factoring by Grouping A. Factor: x3 – x2– 25x + 25. Group terms. Factor common monomials from each group. Factor out the common binomial (x – 1). Factor the difference of squares. Check Use the table feature of your calculator to compare the original expression and the factored form.

  3. Example 2B: Factoring by Grouping B. Factor: x3 – 2x2– 9x + 18. Example 3A: Factoring the Sum or Difference of Two Cubes A. Factor the expression. 4x4 + 108x Factor out the GCF, _____. Rewrite as the sum of cubes. Use the rule a3+ b3 = Example 3B: Factoring the Sum or Difference of Two Cubes B. Factor the expression. 125d3 – 8 Rewrite as the difference of cubes. Use the rule a3– b3 =

  4. Example 3C: Factoring the Sum or Difference of Two Cubes C. Factor the expression. 8 + z6 Example 4: Geometry Application The volume of a plastic storage box is modeled by the function V(x) = x3 + 6x2 + 3x – 10. Identify the values of x for which V(x) = 0, then use the graph to factor V(x). V(x) has _____ real zeros at x =_______________. If the model is accurate, the box will have no volume if x =________________. One corresponding factor is (x _____). Use synthetic division to factor the polynomial. V(x)= (x – 1)(________________) Write V(x) as a product. Factor the quadratic.

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