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Higher-Degree Polynomial Functions and Graphs Ernesto Diaz Professor of Mathematics Copyright © 2011 Pearson Education, Inc. Slide 1.4-1 Section 4.1 Polynomial Functions and Modeling Copyright ©2013, 2009, 2006, 2005 Pearson Education, Inc. Objectives Determine the behavior of the graph of a polynomial function using the leading-term test. Factor polynomial functions and find the zeros and their multiplicities. Use a graphing calculator to graph a polynomial function and find its real-number zeros. Solve applied problems using polynomial models; fit linear, quadratic, power, cubic, and quartic polynomial functions to data. Polynomial Function A polynomial function P is given by P( x ) an x n an1x n1 an2 x n2 ... a1x a0, where the coefficients an, an - 1, …, a1, a0 are real numbers and the exponents are whole numbers. Quadratic Function Cubic Function Examples of Polynomial Functions Examples of Nonpolynomial Functions Polynomial Functions The graph of a polynomial function is continuous and smooth. The domain of a polynomial function is the set of all real numbers. The Leading-Term Test Example Using the leading term-test, match each of the following functions with one of the graphs AD, which follow. f ( x) 3x 4 2 x 3 3 a) b) f ( x) x5 14 x 1 c) f ( x) 5 x 3 x 2 4 x 2 f ( x) x 6 x 5 4 x 3 d) Graphs a. f ( x) 3x 4 2 x3 3 b. f ( x) 5 x3 x 2 4 x 2 c. f ( x) x5 14 x 1 d. f ( x) x 6 x5 4 x3 Solution Leading Term Degree of Leading Term Sign of Leading Coeff. Graph a) 3x4 Even Positive D b) 5x3 Odd Negative B c) x5 Odd Positive A d) x6 Even Negative C Graphs f ( x) 5 x 3 x 2 4 x 2 f ( x) x5 14 x 1 f ( x) x x 4 x 6 5 3 f ( x) 3x 4 2 x 3 3 Finding Zeros of Factored Polynomial Functions If c is a real zero of a function (that is, f(c) = 0), then (c, 0) is an x-intercept of the graph of the function. Example Find the zeros of f ( x) 5( x 1)( x 1)( x 1)( x 2) 5( x 1)3 ( x 2). To solve the equation f(x) = 0, we use the principle of zero products, solving x 1 = 0 and x + 2 = 0. The zeros of f(x) are 1 and 2. See graph on right. Even and Odd Multiplicity If (x c)k, k 1, is a factor of a polynomial function P(x) and (x c)k + 1 is not a factor and: k is odd, then the graph crosses the x-axis at (c, 0); k is even, then the graph is tangent to the x-axis at (c, 0). Example Find the zeros of f(x) = x3 – 2x2 – 9x + 18. Solution We factor by grouping. f(x) = x3 – 2x2 – 9x + 18 = x2(x – 2) – 9(x – 2). ( x 2)( x 2 9) ( x 2)( x 3)( x 3) By the principle of zero products, the solutions of the equation f(x) = 0, are 2, –3, and 3. Example Find the zeros of f(x) = x4 + 8x2 – 33. We factor as follows: f(x) = x4 + 8x2 – 33 = (x2 + 11)(x2 – 3). Solve the equation f(x) = 0 to determine the zeros. We use the principle of zero products. ( x 2 11)( x 2 3) 0 x 2 11 0 x 2 11 or x2 3 0 or x2 3 x 11 or x i 11 x2 3 Example Find the zeros of f(x) = 0.2x3 – 1.5x2 – 0.3x + 2. Approximate the zeros to three decimal places. Solution Use a graphing calculator to create a graph. Look for points where the graph crosses the x-axis. We use the ZERO feature to find them. –10 10 The zeros are approximately –1.164, 1,142, and 7.523. Example The polynomial function M (t ) 0.5t 4 3.45t 3 96.65t 2 347.7t can be used to estimate the number of milligrams of the pain relief medication ibuprofen in the bloodstream t hours after 400 mg of the medication has been taken. Find the number of milligrams in the bloodstream at t = 0, 0.5, 1, 1.5, and so on, up to 6 hr. Round the function values to the nearest tenth. Solution Using a calculator, we compute the function values. Example-continued Using a calculator, we compute the function values. M (0) 0 M (3.5) 255.9 M (0.5) 150.2 M (4) 193.2 M (1) 255 M (4.5) 126.9 M (1.5) 318.3 M (5) 66 M (2) 344.4 M (5.5) 20.2 M (2.5) 338.6 M (6) 0 M (3) 306.9