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2 What You Should Learn Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions. Find rational zeros of polynomial functions. Find conjugate pairs of complex zeros. Find zeros of polynomials by factoring. Use Descartes s Rule of Signs and the Upper and Lower Bound Rules to find zeros of polynomials. 2

3 The Fundamental Theorem of Algebra 3

4 The Fundamental Theorem of Algebra 4

5 Example 1 Zeros of Polynomial Functions a. The first-degree polynomial f (x) = x 2 has exactly one zero: x = 2. b. Counting multiplicity, the second-degree polynomial function f (x) = x 2 6x + 9 = (x 3)(x 3) has exactly two zeros: x = 3 and x = 3 (repeated zero) 5

6 Example 1 Zeros of Polynomial Functions cont d c. The third-degree polynomial function f (x) = x 3 + 4x = x(x 2 + 4) = x(x 2i )(x + 2i ) has exactly three zeros: x = 0, x = 2i, and x = 2i. 6

7 Example 1 Zeros of Polynomial Functions cont d d. The fourth-degree polynomial function f (x) = x 4 1 = (x 1)(x + 1)(x i )(x + i ) has exactly four zeros: x = 1, x = 1, x = i, and x = i. 7

8 The Rational Zero Test 8

9 The Rational Zero Test 9

10 Example 3 Rational Zero Test with Leading Coefficient of 1 Find the rational zeros of f (x) = x 4 x 3 + x 2 3x 6. Solution: Because the leading coefficient is 1, the possible rational zeros are the factors of the constant term. Possible rational zeros: ±1, ±2, ±3, ±6 10

11 Example 3 Solution cont d 0 remainder, so x = 1 is a zero. 0 remainder, so x = 2 is a zero. So, f (x) factors as f (x) = (x + 1)(x 2)(x2 + 3). 11

12 Example 3 Solution cont d Figure

13 The Rational Zero Test If the leading coefficient of a polynomial is not 1, (1) * a programmable calculator (2) a graph:good estimate of the locations of the zeros; (3) the Intermediate Value Theorem along with a table (4) synthetic division can be used to test the possible rational zeros. 13

14 Example Find all the real zeros of f ( x) 10x 15x 16x 12 14

15 Conjugate Pairs 15

16 Conjugate Pairs The pairs of complex zeros of the form a + bi and a bi are conjugates. 16

17 Example 7 Finding the Zeros of a Polynomial Functions Find all the zeros of f ( x) x 4 3x 3 6x 2 2x 60 given that 1 3i is a zero of f. 17

18 Example 6 Finding a Polynomial with Given Zeros Find a fourth-degree polynomial function with real coefficients that has 1, 1, and 3i as zeros. Solution: Because 3i is a zero and the polynomial is stated to have real coefficients, conjugate 3i must also be a zero. So, from the Linear Factorization Theorem, f (x) can be written as f (x) = a (x + 1)(x + 1)(x 3i )(x + 3i ). 18

19 Example 6 Solution cont d For simplicity, let a = 1 to obtain f (x) = (x2 + 2x + 1)(x 2 + 9) = x 4 + 2x x x

20 Factoring a Polynomial 20

21 Factoring a Polynomial f (x) = a n (x c 1 )(x c 2 )(x c 3 )... (x c n ) 21

22 Factoring a Polynomial A quadratic factor with no real zeros is said to be prime or irreducible over the reals. x = (x i )(x + i ) is irreducible over the reals (and therefore over the rationals). x 2 2 = but reducible over the reals. is irreducible over the rationals 22

23 Example 7 Finding the Zeros of a Polynomial Function Find all the zeros of f (x) = x 4 3x 3 + 6x 2 + 2x 60 given that 1 + 3i is a zero of f. Solution: Because complex zeros occur in conjugate pairs, you know that 1 3i is also a zero of f. Both [x (1 + 3i )] and [x (1 3i )] are factors of f. [x (1 + 3i )][x (1 3i )] = [(x 1) 3i ][(x 1) + 3i ] = (x 1) 2 9i 2 = x 2 2x

24 Example 7 Solution cont d 24

25 Example 7 Solution cont d f (x) = (x2 2x + 10)(x2 x 6) = (x 2 2x + 10)(x 3)(x + 2) The zeros of f are x = 1 + 3i, x = 1 3i, x = 3, and x = 2. 25

26 Other Tests for Zeros of Polynomials 26

27 Other Tests for Zeros of Polynomials A variation in sign means that two consecutive coefficients have opposite signs. 27

28 Other Tests for Zeros of Polynomials When using Descartes s Rule of Signs, a zero of multiplicity k should be counted as k zeros. x 3 3x + 2 = (x 1)(x 1)(x + 2) the two positive real zeros are x = 1 of multiplicity 2. 28

29 Example 9 Using Descartes s Rule of Signs Describe the possible real zeros of f (x) = 3x 3 5x2 + 6x 4. Solution: The original polynomial has three variations in sign. f (x) = 3x 3 5x2 + 6x 4. 29

30 Example 9 Solution cont d The polynomial f ( x) = 3 ( x) 3 5 ( x) ( x) 4 = 3x 3 5x 2 6x 4 has no variations in sign. So, from Descartes s Rule of Signs, the polynomial f (x) = 3x 3 5x2 + 6x 4 has either three positive real zeros or one positive real zero, and has no negative real zeros. 30

31 Example 9 Solution cont d 31

32 Other Tests for Zeros of Polynomials 32

33 Other Tests for Zeros of Polynomials 1. If the terms of f (x) have a common monomial factor, it should be factored out before applying the tests in this section. For instance, by writing f (x) = x4 5x 3 + 3x2 + x = x(x 3 5x 2 + 3x + 1) you can see that x = 0 is a zero of f and that the remaining zeros can be obtained by analyzing the cubic factor. 33

34 Other Tests for Zeros of Polynomials 2. If you are able to find all but two zeros of f (x), you can always use the Quadratic Formula on the remaining quadratic factor. For instance, if you succeeded in writing f (x) = x4 5x 3 + 3x2 + x = x(x 1)(x 2 4x 1) you can apply the Quadratic Formula to x 2 4x 1 to conclude that the two remaining zeros are x = 2 + x = 2. 34

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