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Copyright © 2007 Pearson Education, Inc. Slide 3-1 Chapter 3: Polynomial Functions 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 Complex Numbers Quadratic Functions and Graphs Quadratic Equations and Inequalities Further Applications of Quadratic Functions and Models Higher Degree Polynomial Functions and Graphs Topics in the Theory of Polynomial Functions (I) Topics in the Theory of Polynomial Functions (II) Polynomial Equations and Inequalities; Further Applications and Models Copyright © 2007 Pearson Education, Inc. Slide 3-2 3.7 Topics in the Theory of Polynomial Functions (II) Complex Zeros and the Fundamental Theorem of Algebra • It can be shown that if a + bi is a zero of a polynomial function with real coefficients, then so is its complex conjugate, a – bi. Conjugate Zeros Theorem If P(x) is a polynomial having only real coefficients, and if a + bi is a zero of P, then the conjugate a – bi is also a zero of P. Copyright © 2007 Pearson Education, Inc. Slide 3-3 3.7 Topics in the Theory of Polynomial Functions (II) Example Find a polynomial having zeros 3 and 2 + i that satisfies the requirement P(–2) = 4. Solution Since 2 + i is a zero, so is 2 – i. A general solution is P ( x ) a ( x 3)[ x ( 2 i )][ x ( 2 i )] a ( x 3)( x 2 i )( x 2 i ) a ( x 7 x 17 x 15). 3 2 Since P(–2) = 4, we have 4 4 a[(2) 7( 2) 17( 2) 15] a . 85 4 3 4 3 28 2 4 12 2 P ( x ) ( x 7 x 17 x 15) x x x . 85 85 85 5 17 3 Copyright © 2007 Pearson Education, Inc. 2 Slide 3-4 3.7 The Fundamental Theorem of Algebra The Fundamental Theorem of Algebra Every function defined by a polynomial of degree 1 or more has at least one complex (real or imaginary) zero. • If P(x) is a polynomial of degree 1 or more, then there is some number k such that P(k) = 0. In other words, P( x) ( x k ) Q( x), where Q(x) can also be factored resulting in P( x) a( x k1 )( x k2 )( x kn ). Copyright © 2007 Pearson Education, Inc. Slide 3-5 3.7 Zeros of a Polynomial Function Number of Zeros Theorem A function defined by a polynomial of degree n has at most n distinct complex zeros. Example Find all complex zeros of P( x) x 4 7 x3 18x 2 22 x 12 given that 1 – i is a zero. Solution 1 i 1 1 Copyright © 2007 Pearson Education, Inc. 7 18 22 12 1 i 7 5i 16 6i 12 6 i 11 5i 6 6i 0 Slide 3-6 3.7 Zeros of a Polynomial Function Using the Conjugate Zeros Theorem, 1 + i is also a zero. 1 i 1 1 6 i 11 5i 1 i 5 5i 5 6 6 6i 6 6i 0 The zeros of x2 – 5x + 6 are 2 and 3. Thus, P( x) x 4 7 x 3 18 x 2 22 x 12 ( x 2)( x 3)( x 1 i)( x 1 i ) and has four zeros: 1 – i, 1 + i, 2, and 3. Copyright © 2007 Pearson Education, Inc. Slide 3-7 3.7 Multiplicity of a Zero • The multiplicity of the zero refers to the number of times a zero appears. P( x) x 6 x 5 5 x 4 x 3 8 x 2 4 x x( x 2) 2 ( x 1)3 e.g. – – – x = 0 leads to a single zero (x + 2)2 leads to a zero of –2 with multiplicity two (x – 1)3 leads to a zero of 1 with multiplicity three Copyright © 2007 Pearson Education, Inc. Slide 3-8 3.7 Polynomial Function Satisfying Given Conditions Example Find a polynomial function with real coefficients of lowest possible degree having a zero 2 of multiplicity 3, a zero 0 of multiplicity 2, and a zero i of single multiplicity. By the conjugate zeros theorem, –i is also a zero. Solution P( x) x 2 ( x 2)3 ( x i )( x i ) x 7 6 x 6 13x 5 14 x 4 12 x 3 8 x 2 This is one of many such functions. Multiplying P(x) by any nonzero number will yield another function satisfying these conditions. Copyright © 2007 Pearson Education, Inc. Slide 3-9 3.7 Observation: Parity of Multiplicities of Zeros Observe the behavior around the zeros of the polynomials P( x) ( x 3)( x 2) 2 P( x) ( x 3) 2 ( x 2)3 P( x) x 2 ( x 1)( x 2) 2 . The following figure illustrates some conclusions. By observing the dominating term and noting the parity of multiplicities of zeros of a polynomial in factored form, we can draw a rough graph of a polynomial by hand. Copyright © 2007 Pearson Education, Inc. Slide 3-10 3.7 Sketching a Polynomial Function by Hand Example Sketch P( x) 2( x 4) 2 ( x 3)( x 1) 2 by hand. Solution The dominating term is –2x5, so the end behavior will rise on the left and fall on the right. Because –4 and 1 are x-intercepts determined by zeros of even multiplicity, the graph will be tangent to the x-axis at these x-intercepts. The y-intercept is –96. Copyright © 2007 Pearson Education, Inc. Slide 3-11 3.7 The Rational Zeros Theorem The Rational Zeros Theorem Let P( x) an x n an 1 x n 1 a1 x a0 , where an 0, define a polynomial function w ith integer coefficients. If p / q is a rational number wri tten in lowest terms, and if p / q is a zero of P, then p is a factor of the constant term a0 , and q is a factor of the leading coefficient an . Example P( x) 6 x 4 7 x3 12 x 2 3x 2 (a) List all possible rational zeros. (b) Use a graph to eliminate some of the possible zeros listed in part (a). (c) Find all rational zeros and factor P(x). Copyright © 2007 Pearson Education, Inc. Slide 3-12 3.7 The Rational Zeros Theorem Solution (a) a0 2 with factors 1, 2; a4 6 with factors 1, 2, 3, 6; p 1 1 1 2 possible rational zeros are 1, 2, , , , q 2 3 6 3 (b) From the graph, the zeros are no less than –2 and no greater than 1. Also, –1 is clearly not a zero since the graph does not intersect the x-axis at the point (-1,0). Copyright © 2007 Pearson Education, Inc. Slide 3-13 3.7 The Rational Zeros Theorem (c) Show that 1 and –2 are zeros. 16 6 2 6 6 7 6 13 12 13 1 13 1 12 2 1 1 3 1 2 2 2 0 2 2 0 Solving the equation 6x2 + x – 1 = 0, we get x = –1/2, 1/3. 1 1 P( x) 6( x 1)( x 2) x x 3 2 ( x 1)( x 2)(3x 1)(2 x 1) Copyright © 2007 Pearson Education, Inc. Slide 3-14 3.7 Descartes’ Rule of Sign Let P(x) define a polynomial function with real coefficients and a nonzero constant term, with terms in descending powers of x. (a) The number of positive real zeros of P either equals the number of variations in sign occurring in the coefficients of P(x) or is less than the number of variations by a positive even integer. (b) The number of negative real zeros of P is either the number of variations in sign occurring in the coefficients of P(x) or is less than the number of variations by a positive even integer. Copyright © 2007 Pearson Education, Inc. Slide 3-15 3.7 Applying Descartes’ Rule of Signs Example Determine the possible number of positive real zeros and negative real zeros of P(x) = x4 – 6x3 + 8x2 + 2x – 1. We first consider the possible number of positive zeros by observing that P(x) has three variations in signs. + x4 – 6x3 + 8x2 + 2x – 1 1 2 3 Thus, by Descartes’ rule of signs, f has either 3 or 3 – 2 = 1 positive real zeros. For negative zeros, consider the variations in signs for P(x). P(x) = (x)4 – 6(x)3 + 8(x)2 + 2(x) 1 = x4 + 6x3 + 8x2 – 2x – 1 Since there is only one variation in sign, P(x) has only one negative real root. Copyright © 2007 Pearson Education, Inc. Slide 3-16 3.7 Boundedness Theorem Let P(x) define a polynomial function of degree n 1 with real coefficients and with a positive leading coefficient. If P(x) is divided synthetically by x – c, and (a) if c > 0 and all numbers in the bottom row of the synthetic division are nonnegative, then P(x) has no zero greater than c; (b) if c < 0 and the numbers in the bottom row of the synthetic division alternate in sign (with 0 considered positive or negative, as needed), then P(x) has no zero less than c. Copyright © 2007 Pearson Education, Inc. Slide 3-17 3.7 Using the Boundedness Theorem Example Show that the real zeros of P(x) = 2x4 – 5x3 + 3x + 1 satisfy the following conditions. (a) No real zero is greater than 3. (b) No real zero is less than –1. Solution a) c > 0 3 2 5 0 3 1 6 3 9 36 2 1 3 12 37 All are nonegative. No real zero greater than 3. b) c < 0 1 2 5 0 3 1 6 7 7 4 2 7 7 4 5 Copyright © 2007 Pearson Education, Inc. The numbers alternate in sign. No zero less than 1. Slide 3-18