1. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Real Zeros of Polynomial Functions ♦ Divide Polynomials ♦ Understand the division algorithm, remainder theorem, and factor theorem ♦ Factor higher degree polynomials ♦ Analyze polynomials with multiple zeros ♦ Find rational zeros ♦ Solve polynomial equations 4.3

2. Slide 4- 2 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example-Dividing by a monomial Divide 6x 3 - 3x 2 +2 by 2x 2. Check the result. Solution

3. Slide 4- 3 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Divide x 3 + 2x 2  5x  6 by x  3. Check the result. Solution

4. Slide 4- 4 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution continued Check

5. Slide 4- 5 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Synthetic Division A short cut called synthetic division can be used to divide x – k into a polynomial. Steps 1. Write k to the left and the coefficients of f(x) to the right in the top row. If any power does not appear in f(x), include a 0 for that term. 2. Copy the leading coefficient of f(x) into the third row and multiply it by k. Write the result below the next coefficient of f(x) in the second row. Add the numbers in the second column and place the result in the third row. Repeat the process. 3. The last number in the third row is the remainder. If the remainder is 0, then the binomial x – k is a factor of f(x). The other numbers in the third row are the coefficients of the quotient in descending powers.

6. Slide 4- 6 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Use synthetic division to divide 2x 3 + 7x 2 – 5 by x + 3.

7. Slide 4- 7 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

8. Slide 4- 8 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

9. Slide 4- 9 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Use the graph of f(x) = x 3  x 2 – 9x + 9 and the factor theorem to list the factors of f(x). Solution

10. Slide 4- 10 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

11. Slide 4- 11 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Write the complete factorization for the polynomial 6x 3 + 19x 2 + 2x – 3 with given zeros –3, –1/2 and 1/3. Solution

12. Slide 4- 12 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example The polynomial f(x) = 2x 3  3x 2  17x + 30 has a zero of 2. Express f(x) in complete factored form. Solution

13. Slide 4- 13 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Rational Zeros

14. Slide 4- 14 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Find all rational zeros of f(x) = 6x 4 + 7x 3  12x 2  3x + 2. Write in complete factored form. Solution Any rational zero must occur in the list

15. Slide 4- 15 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution continued Evaluate f(x) at each value in the list. xf(x)f(x)Xf(x)f(x)xf(x)f(x) 1½1/61.20 11 ½½  1/6 2.14 21/32/3  2.07 22  1/3 1.48  2/3  2.22

16. Slide 4- 16 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Find all real solutions to the equation 4x 4 – 17x 2 – 50 = 0. Solution

17. Slide 4- 17 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Solve the equation x 3 – 2.1x 2 – 7.1x + 0.9 = 0 graphically. Round any solutions to the nearest hundredth. Solution