SECTION 4.4 Complex Zeros; Fundamental Theorem of Algebra 229 123. Challenge Problem Use the Intermediate Value Theorem to show that the functions y x3 = and y x 1 2 = − intersect somewhere between x 0 = and x 1 = . 122. Challenge Problem Prove the Rational Zeros Theorem. [ Hint : Let p q , where p and q have no common factors except 1 and 1, − be a zero of the polynomial function f x a x a x a x a n n n n 1 1 1 0 ( ) = + + + + − − whose coefficients are all integers. Show that a p a p q a pq a q 0 n n n n n n 1 1 1 1 0 + + + + = − − − Now, show that p must be a factor of a0 , and that q must be a factor of an .] Problems 128–137 are based on previously learned material. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for later sections, a final exam, or subsequent courses such as calculus. Retain Your Knowledge 128. Write f x x x 3 30 4 2 ( ) = − + − in the form f x a x h k 2 ( ) ( ) = − + 129. Express the inequality x 3 8 ≤ < using interval notation. 130. Solve x y y 2 5 3 for − = . 131. Solve x x 1 3 2 9 0 2 − + = . x y 6 6 (–3, –2) (–1, 0) (–5, 0) (0, 3) (2, 6) (5, 1) y = f (x) –6 For Problems 132–137, use the graph on the right. 132. On which interval(s) is f increasing? 133. On which interval(s) is f decreasing? 134. What are the zeros of f , if any? 135. What are the intercepts of the graph of f ? 136. What are the turning points? 137. What are the absolute extrema, if any? ‘Are You Prepared?’ Answers 1. 3 2. x x 3 2 2 1 ( )( ) + − 3. Quotient: x x x 3 4 12 43; 3 2 + + + Remainder: 125 4. 1 13 2 , 1 13 2 − − − + Explaining Concepts 124. Is 1 3 a zero of f x x x x 2 3 6 7? 3 2 ( ) = + − + Explain. 125. Is 3 5 a zero of ( ) = − + − + f x x x x x 2 5 1? 6 4 3 Explain. 126. Is 1 3 a zero of f x x x x 4 5 3 1? 3 2 ( ) = − − + Explain. 127. Is 2 3 a zero of ( ) = + − + + f x x x x x 6 2? 7 5 4 Explain. 4.4 Complex Zeros; Fundamental Theorem of Algebra Now Work the ‘Are You Prepared?’ problems on page 234. • Complex Numbers (Section A.7, pp. A58–A63) • Complex Solutions of a Quadratic Equation (Section A.7, pp. A63–A65) PREPARING FOR THIS SECTION Before getting started, review the following: OBJECTIVES 1 Use the Conjugate Pairs Theorem (p. 231) 2 Find a Polynomial Function with Specified Zeros (p. 232) 3 Find the Complex Zeros of a Polynomial Function (p. 233) In Section A.6, we found the real solutions of a quadratic equation.That is, we found the real zeros of a polynomial function of degree 2. Then, in Section A.7 we found the complex solutions of a quadratic equation. That is, we found the complex zeros of a polynomial function of degree 2.
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