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6.5 Warm Up

6.5 Warm Up. Factor 8x 3 + 125. Factor 5x 3 + 10x 2 – x – 2. Factor 200x 6 – 2x 4 . Find the product of (2x – 3)(2x – 5). Find the product of (5x + 2y) 3. 6.5 Polynomial Division. Numerator. Long division. Denominator. Quotient. Remainder expressed as a fraction.

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6.5 Warm Up

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  1. 6.5 Warm Up • Factor 8x3 + 125. • Factor 5x3 + 10x2 – x – 2. • Factor 200x6 – 2x4. • Find the product of (2x – 3)(2x – 5). • Find the product of (5x + 2y)3

  2. 6.5 Polynomial Division Numerator Long division Denominator Quotient Remainder expressed as a fraction.

  3. Long Division Example Remainder

  4. Long Division Example 2 Notice x2 –1 is missing a term. Remainder

  5. Classwork Textbook page 356 probs 4 – 7.

  6. Synthetic Division Synthetic division can be used to divide polynomials by an expression in the form of x - k. Example: Divide (x3 – 8x + 3) by (x + 3). x + 3 is in the form of x – k. x + 3 = 0 x = -3 1 0 -8 3 -3 -3 9 -3 1 -3 1 0 Remainder x2 - 3x + 1 Quotient

  7. Classwork Textbook page 356 probs 8 - 11.

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

  9. -3 1 0 -8 3 -3 9 -3 1 -3 1 0 Synthetic Division Continued Example: Factor (x3 – 8x + 3) given (x + 3) is a factor. Continue factoring x2 - 3x + 1 So,

  10. Synthetic Division Example 2 Given one zero of the polynomial function, find the other zeros. 10 2 -14 –56 -40 20 60 40 2 6 4 0 2x2 + 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 = -1 -1 3 8 -45 -50 x + 1 50 -3 -5 0 3 5 -50 ? x + 5 x = -5 -5 3 5 -50 -15 50 0 3 -10 3x -10

  13. Classwork Textbook page 356 probs 12 - 13. Homework Textbook page 356 probs 15 – 45 1st col.

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