1. 6.5 Warm Up 1.Factor 8x 3 + 125. 2.Factor 5x 3 + 10x 2 – x – 2. 3.Factor 200x 6 – 2x 4. 4.Find the product of (2x – 3)(2x – 5). 5.Find the product of (5x + 2y) 3

2. 6.5 Polynomial Division Long division Numerator DenominatorQuotientRemainder expressed as a fraction.

3. Long Division Example Remainder

4. Long Division Example 2 Notice x 2 –1 is missing a term. Remainder

5. Classwork Workbook page 115 probs 1-4

6. Synthetic Division Synthetic division can be used to divide polynomials by an expression in the form of x - k. Example: Divide (x 3 – 8x + 3) by (x + 3). x + 3 is in the form of x – k. x + 3 = 0 x = -3 -3 1 0 -8 3 1 -3 9 1 0 Remainder x 2 - 3x + 1 Quotient

7. Classwork Workbook page 116 probs 5-9.

8. Remainder and Factor Theorem Remainder Theorem If a polynomial f(x) is divided by x – k, then the remainder is r = f(k). Factor Theorem A polynomial f(x) has a factor x – k if and only if f(k) = 0. Example: Problem 7 page 116 had a remainder of –7. f(-4) = the remainder.

9. Synthetic Division Continued Example: Factor (x 3 – 8x + 3) given (x + 3) is a factor. x 2 - 3x + 1 1 0 -8 3 1 -3 9 1 0 Continue factoring So,

10. Synthetic Division Example 2 Given one zero of the polynomial function, find the other zeros. 10 2 -14 –56 -40 2 20 6 60 4 40 0 2x 2 + 6x + 4 (2x + 4)(x + 1) (2x + 4) = 0 (x + 1) = 0 x = -2 x = -1 So, the zeros of the polynomial are 10, -1, and –2.

11. A Geometric Interpretation The real zeros are the x-intercepts of the graph.

12. A Geometry Problem Given the expression for the volume of a rectangular prism, find an expression for the missing dimension. x + 5 x + 1 ? x = -1 -1 3 8 -45 -50 3 -3-5 5-50 50 0 x = -5 -5 3 5 -50 3 -15 -10 50 0 3x -10

13. Classwork Workbook page 117 problems 9, 10.