1. College Algebra Chapter 3 Polynomial and Rational Functions
Section 3.4
Zeros of Polynomials
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2. Concepts Apply the Rational Zero Theorem
Apply the Fundamental Theorem of Algebra
Apply Descartes’ Rule of Signs
Find Upper and Lower Bounds

3. Concept 1 Apply the Rational Zero Theorem

4. Apply the Rational Zero Theorem (1 of 2)

5. Apply the Rational Zero Theorem (2 of 2)
If a rational zero exists for a polynomial, then it must be of the form:

6. Example 1
List all possible rational zeros of

7. Skill Practice 1
List all possible rational zeros.

8. Example 2
Find the zeros and their multiplicities.

9. Example 3
Find the zeros and their multiplicities.

10. Skill Practice 2
Find the zeros.

11. Skill Practice 3
Find the zeros and their multiplicities.

12. Skill Practice 4
Find the zeros.

13. Concept 2 Apply the Fundamental Theorem of Algebra

14. Apply the Fundamental Theorem of Algebra (1 of 2)
Fundamental Theorem of Algebra: If f(x) is a polynomial of degree n ≥ 1 with complex coefficients, then f(x) has at least one complex zero. Linear Factorization Theorem:

15. Apply the Fundamental Theorem of Algebra (2 of 2)
Conjugate Zeros Theorem: If f(x) is a polynomial with real coefficients and if a + bi (b ≠ 0) is a zero of f (x) then its conjugate a – bi is also a zero of f(x). Number of Zeros of a Polynomial: If f (x) is a polynomial of degree n ≥ 1 with complex coefficients, then f (x) has exactly n complex zeros provided that each zero is counted by its multiplicity.

16. Example 4
Given that 1-5i is a zero of f(x). Find the remaining zeros and factor f(x) as a product of linear factors.

17. Example 5

18. Skill Practice 5 and that 1 + 5i is a zero of f(x), Find the zeros.
Factor f(x) as a product of linear factors.
Solve the equation.

19. Example 6
Find a polynomial f(x) of lowest degree with zeros of 3i and 2 (multiplicity 2).

20. Example 7
Find a polynomial p(x) of degree 3 with zeros – 2i and 4. The polynomial must also satisfy the condition that p(0) = 32.

21. Skill Practice 6
Find a third-degree polynomial f(x) with integer coefficients and with zeros of 2 + i and
Find a polynomial g(x) of lowest degree with zeros of -3 (multiplicity 2) and 5 (multiplicity 2), and satisfying the condition that g(0) = 450.

22. Concept 3 Apply Descartes’ Rule of Signs

23. Apply Descartes’ Rule of Signs
Descartes' Rule of Signs: Let f (x) be a polynomial with real coefficients and a nonzero constant term. Then, The number of positive real zeros is either the same as the number of sign changes in f(x) or less than the number of sign changes in f(x) by a positive even integer. The number of negative real zeros is either the same as the number of sign changes in f(–x) or less than the number of sign changes in f(–x) by a positive even integer.

24. Example 8
Determine the number of possible positive and negative real zeros.
Number of possible positive real zeros
Number of possible negative real zeros
Number of imaginary zeros
Total (including multiplicities)

25. Skill Practice 7
Determine the number of possible positive and negative real zeros.

26. Skill Practice 8
Determine the number of possible positive and negative real zeros.

27. Concept 4 Find Upper and Lower Bounds

28. Find Upper and Lower Bounds (1 of 2)
A real number b is called an upper bound of the real zeros of a polynomial if all real zeros are less than or equal to b. A real number a is called a lower bound of the real zeros of a polynomial if all real zeros are greater than or equal to a.

29. Find Upper and Lower Bounds (2 of 2)
Let f (x) be a polynomial of degree n ≥ 1 with real coefficients and a positive leading coefficient. Further suppose that f (x) is divided by (x – c).
If c > 0 and if both the remainder and the coefficients of the quotient are nonnegative, then c is an upper bound for the real zeros of f (x).
If c < 0 and the coefficients of the quotient and the remainder alternate in sign (with 0 being considered either positive or negative as needed), then c is a lower bound for the real zeros of f.

30. Example 9
Given p(x), show that 4 is an upper bound and –2 is a lower bound for the real zeros.

31. Skill Practice 9
Determine if the upper bound theorem identifies 4 as an upper bound for the real zeros of f(x).
Determine if the lower bound theorem identifies -4 as a lower bound for the real zeros of f(x).

32. Skill Practice 10
Find the zeros and their multiplicities.