266 CHAPTER 4 Polynomial and Rational Functions Chapter Review Things to Know Power function (pp. 191–193) ( ) = ≥ f x x n , 2 n even Domain: all real numbers Range: nonnegative real numbers Contains the points ( ) ( ) ( ) −1, 1 , 0, 0 , 1, 1 Even function Decreasing on ( ] −∞, 0 , increasing on [ )∞ 0, ( ) = ≥ f x x n , 3 n odd Domain: all real numbers Range: all real numbers Contains the points ( ) ( ) ( ) − − 1, 1, 0,0, 1,1 Odd function Increasing on ( ) −∞ ∞, Polynomial function (pp. 190–201) Domain: all real numbers ( ) = + − − f x a x a x n n n n 1 1 At most −n 1 turning points + + + ≠ a x a a , 0 n 1 0 End behavior: Behaves like = y a x x for large n n Real zeros of a polynomial function f (p. 195) Real numbers for which ( ) = f x 0; the real zeros of f are the x-intercepts of the graph of f. Multiplicity (p. 196) If ( ) −x r m is a factor of a polynomial f and ( ) − + x r m 1 is not a factor of f, then r is called a real zero of multiplicity m of f . Remainder Theorem (p. 215) If a polynomial function ( ) f x is divided by −x c, then the remainder is ( ) f c . Factor Theorem (p. 216) −x c is a factor of a polynomial function ( ) f x if and only if ( ) = f c 0. Descartes’ Rule of Signs (p. 217) Let f denote a polynomial function written in standard form. • The number of positive real zeros of f either equals the number of variations in the sign of the nonzero coefficients of ( ) f x or else equals that number less an even integer. • The number of negative real zeros of f either equals the number of variations in the sign of the nonzero coefficients of ( ) − f x or else equals that number less an even integer. Rational Zeros Theorem (p. 218) Let f be a polynomial function of degree 1 or higher of the form ( ) = + + + + ≠ ≠ − − f x a x a x a x a a a 0, 0 n n n n n 1 1 1 0 0 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. Intermediate Value Theorem (p. 225) Let f be a polynomial function. If < a b and ( ) f a and ( ) f b are of opposite sign, then there is at least one real zero of f between a and b. Fundamental Theorem of Algebra (p. 230) Every complex polynomial function f of degree ≥ n 1 has at least one complex zero. Conjugate Pairs Theorem (p. 231) Let f be a polynomial function whose coefficients are real numbers. If = + r a bi is a zero of f, then its complex conjugate = − r a bi is also a zero of f. Rational function (pp. 236–244) ( ) ( ) ( ) = R x p x q x Domain: { } ( ) ≠ x q x 0 p, q are polynomial functions and q is not the zero polynomial. Vertical asymptotes: With ( ) R x in lowest terms, if ( ) = q r 0 for some real number, then = x r is a vertical asymptote. Horizontal or oblique asymptote: See the summary on page 242. Objectives Section You should be able to . . . Examples Review Exercises 4.1 1 Identify polynomial functions and their degree (p. 190) 1 1–4 2 Graph polynomial functions using transformations (p. 194) 2, 3 5–7 3 Identify the real zeros of a polynomial function and their multiplicity (p. 194) 4–9 8–11
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