Sullivan Algebra and Trigonometry: Section 5.5 Real Zeros of a Polynomial Function

1. Sullivan Algebra and Trigonometry: Section 5.5Real Zeros of a Polynomial Function • Objectives • Use the Remainder and Factor Theorems • Use the Rational Zeros Theorem • Find the Real Zeros of a Polynomial Function • Solve Polynomial Equations • Use the Intermediate Value Theorem

2. Division Algorithm for Polynomials If f(x) and g(x) denote polynomial functions and if g(x) is not the zero polynomial, then there are unique polynomial functions q(x) and r(x) such that dividend quotient divisor remainder

3. 3 2 f ( x ) = 3 x + 9 x - 18 x - 24 Find the remainder if is divided by x + 3. Remainder Theorem Let f be a polynomial function. If f (x) is divided by x - c, then the remainder is f (c). x + 3 = x - (-3)

4. Factor Theorem Let f be a polynomial function. Then x - c is a factor of f (x) if and only if f (c) = 0. In other words, if f(c) = 0, then the remainder found if f(x) is divided by x - c is zero. Hence, since x - c divided into f(x) evenly (remainder 0), x - c is a factor of f(x).

5. Use the Factor Theorem to determine whether the 3 2 f ( x ) = 3 x + 9 x - 18 x - 24 has the factor function (a) x + 3 (b) x + 4 a.) f(-3) = 30 Therefore, x + 3 does not divide into f. So, x + 3 is not a factor of f. b.) f(-4) = 0 Therefore, x + 4 does divide into f. So, x + 4 is a factor of f.

6. Rational Zeros Theorem Let f be a polynomial function of degree 1 or higher of the form where each coefficient is an integer. If p/q, in lowest terms, is a rational zero of f, then p must be a factor of a0 and q must be a factor of an.

7. 3 2 f ( x ) = 2 x + 3 x - 23 x - 12 Factors of p: Factors of q: Potential Rational Zeros List the potential rational zeros of According the the theorem, the numerator of potential rational zeros will be factors of p = - 12 and the denominator will be factors of q = 2

8. 5 4 3 2 f ( x ) = x + x - 9 x - x + 20 x - 12 Find the real zeros of Factor f over the real numbers. First, determine the nature of the zeros. Since the polynomial is degree 5, there are at most five zeros.

9. 5 4 3 2 f ( x ) = x + x - 9 x - x + 20 x - 12 Now, list all possible rational zeros p/q by factoring the first and last coefficients of the function. Now, begin testing each potential zero using synthetic division. If a potential zero k is in fact a zero, then x - k divides into f (remainder will be zero) and is a factor of f.

10. 5 4 3 2 f ( x ) = x + x - 9 x - x + 20 x - 12 Test k = -3 Test k = -2 Thus, -3 is a zero of f and x + 3 is a factor of f. Thus, -2 is a zero of f and x + 2 is a factor of f.

11. Test k = 1 Thus, 1 is a zero of f and x - 1 is a factor of f.

12. 5 4 3 2 f ( x ) = x + x - 9 x - x + 20 x - 12 Theorem: Every polynomial function (with real coefficients) can be uniquely factored into a product of linear factors and/or irreducible quadratic factors. In the previous example, while finding the real zeros of the polynomial, f(x) was factored as follows: The function was factored into four uniquely factors, one of which had multiplicity 2.

13. Intermediate Value Theorem Let f denote a continuous function. If a < b and if f(a) and f(b) are of opposite sign, then the graph of f has at least one x-intercept between a and b. y f(b) x-intercept a x b f(a)

14. 3 2 f ( x ) = x + 0 . 2 x - 5 x - 1 Use the intermediate value theorem to show that the graph of the function has an x - intercept in the interval [-3, -2] f (- 3) = -11.2 < 0 f (-2) = 1.8 > 0 Therefore, since f (-3) < 0 and f (-2) > 0, there exists a number between - 3 and -2 where f (x) = 0. So, the function has an x - intercept in the interval [-3,-2]