Unit 4 Polynomial/Rational Functions Zeros of Polynomial Functions (Unit 4.3) William (Bill) Finch Mathematics Department Denton High School Lesson Goals When you have completed this lesson you will: Find complex solutions to quadratic functions. Apply the Fundamental Theorem of Algebra. Find all complex zeros (real and imaginary) of polynomial functions. Zeros 2 / 14
Fundamental Theorem of Algebra The Fundamental Theorem of Algebra If f (x) is a polynomial of degree n, where n > 0, then f has at least one zero in the complex number system. Linear Factorization Theorem If f (x) is a polynomial of degree n, where n > 0, then f has precisely n linear factors f (x) = a n (x c 1 )(x c 2 ) (x c n ) where c 1, c 2,..., c n are complex numbers. Zeros 3 / 14 Zeros of a Polynomial Function Function Degree Linear Factors Zeros f (x) = x 4 1 x 4 4 g(x) = x 2 4x + 4 2 (x 2)(x 2) 2, 2 h(x) = x 3 + 9x 3 x(x ± 3i) 0, ±3i j(x) = x 4 16 4 (x ± 2)(x ± 2i) ±2, ±2i Degree = # of Linear Factors = # of Zeros Zeros 4 / 14
Finding the Zeros of a Polynomial Function To find the zeros of a polynomial function you may use one of the following tools (or a combination of them): Graphing Factoring Division (long or synthetic) In order to get started it is often helpful to identify a rational zero by either graphing or with the Rational Zero Test. Zeros 5 / 14 Identifying Rational Zeros The Rational Zero Theorem If the polynomial f (x) = a n x n + a n 1 x n 1 + + a 1 x + a 0 has integer coefficients, every rational zero of f has the form Rational zero = p q where p and q have no common factors other than ±1, and p = an integer factor of the constant term a 0 q = an integer factor of the leading coefficient a n Zeros 6 / 14
Example 1 For the polynomial f (x) = x 3 15x 2 + 75x 125 a) List all possible rational zeros. b) Determine which are actually zeros of f. Zeros 7 / 14 Example 2 For the polynomial g(x) = 2x 4 9x 3 18x 2 + 71x 30 a) List all possible rational zeros. b) Determine which are actually zeros of g. Zeros 8 / 14
Complex Zeros Complex Zeros Occur in Conjugate Pairs Let f (x) be a polynomial function that has real coefficients. If a + bi (where b 0) is a zero of the function, the conjugate a bi is also a zero of the function. Zeros 9 / 14 Example 3 Find all of the zeros of f (x) = x 3 4x 2 + 21x 34 if you know one of the zeros is 1 + 4i. Zeros 10 / 14
Example 4 For the polynomial h(x) = 8x 3 4x 2 + 6x 3 a) Find all of the zeros of the function. b) Find all of the linear factors of the function. Zeros 11 / 14 Example 5 Write a third-degree polynomial function whose zeros include 2 and 7i. Zeros 12 / 14
Example 6 The water level in a bucket sitting on a patio can be modeled by f (x) = x 3 + 4x 2 2x + 7 where f (x) is the height of the water in millimeters and x is the time in days. On what day(s) will the water reach a height of 10 millimeters? Zeros 13 / 14 What You Learned You can now: Find complex solutions to quadratic functions. Apply the Fundamental Theorem of Algebra. Find all complex zeros (real and imaginary) of polynomial functions. Do problems Chap 2.4 #1-7 odd, 11, 15, 17, 33, 35, 39, 49-53 odd, 61-65 odd Zeros 14 / 14