1. 3.6 The Real Zeros of Polynomial Functions Goals: Finding zeros of polynomials Factoring polynomials completely

2. Review

3. Review : Synthetic Division

4. 2. Remainder Theorem

5. 2) Determine the remainder of 1) Determine the remainder of 3. Application of Remainder Theorem

6. 4. Recall: Factor Theorem

7. 5. Application of Factor Theorem 1) Is x + 1 a factor of ? 2) Is x - 1 a factor of ?

8. 6. Factoring of Polynomials If yes, then write f(x) in factored form:

9. summary

10. Types of Zeros: Example of a factored polynomial: 6. Real zeros of a polynomial Number of Real Zeros Theorem A polynomial of degree n, has at most n real zeros. Rational Zeros Irrational Zeros Complex Imaginary Zeros

11. 6. Real zeros of a polynomial 2 Methods for finding the zeros 1)Graphing calculator (gives approximation to irrational zeros) 2)Algebraically ( better for finding exact value of zeros)

12. 6 a) graphing calculator approx. Graph: p. 184 #81. x-intercepts: Use ZERO feature y-intercepts: TRACE: x=0 d) Table to determine graph close to zero. Is it above or below? e) Max/Min Find zeros (x-intercepts) using graphing calculator.

13. 6b) Identify Rational Zeros List all possible rational zeros.

14. 8. Test a potential zero 1) graphing calculator (TABLE or Trace) OR 2) Does f(c) = 0 ?

15. Finding both rational and irrational zeros. 9. Determine the zeros of a polynomial 1)Find zeros on calculator and verify f(c) = 0 2)How many zeros (x-intercepts) are there? 3)Are any zeros repeated? 4)Continue synthetic division on previous solution until quotient is factorable. 1)Find zeros on calculator and verify f(c) = 0 2)How many zeros (x-intercepts) are there? 3)Are any zeros repeated? 4)Continue synthetic division on previous solution until quotient is factorable.

16. Example: This function is completely factored 10. Write the complete factorization Write as product of: linear factors and irreducible quadratic factors

17. 10. Write the complete factorization Example 3: 1)Find rational zeros 2)Perform synthetic division on each quotient 3)Repeat until reduced to easily factorable quotient. 1)Find rational zeros 2)Perform synthetic division on each quotient 3)Repeat until reduced to easily factorable quotient.

18. 6. Write the complete factorization Look for repeated zeros (where graph touches at the zero)

19. 6. Write the complete factorization Reduces to difference of squares that can be factored.

20. 6. Write the complete factorization If integer zeros are not found on calculator, look for zeros from list of potential zeros. {p/q} and verify.