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Sullivan Algebra and Trigonometry: Section 5.1 Polynomial Functions Objectives • Identify Polynomials and Their Degree • Graph Polynomial Functions Using Transformations • Identify the Zeros of a Polynomial and Their Multiplicity • Analyze the Graph of a Polynomial Function A polynomial function is a function of the form f ( x ) a n x a n 1 x n n 1 a 1 x a 0 where an , an-1 ,…, a1 , a0 are real numbers and n is a nonnegative integer. The domain consists of all real numbers. The degree of the polynomial is the largest power of x that appears. Example: Determine which of the following are polynomials. For those that are, state the degree. (a) f ( x ) 3x 4 x 5 2 Polynomial of degree 2 (b) h( x ) 3 x 5 Not a polynomial 5 3x (c) F ( x ) 5 2x Not a polynomial Graph the following function using transformations. f ( x ) 4 2 x 1 2( x 1) 4 4 4 15 15 (1,1) 5 (0,0) 0 5 15 yx (0,0) 5 0 (1, -2) 15 4 y 2 x 4 5 15 15 (1, 4) (1,0) 5 0 (2, 2) 5 (2,-2) 5 0 5 15 15 y 2 x 1 4 y 2x 1 4 4 f ( x ) ( x 1)( x 4) Consider the polynomial: 2 Solve the equation f (x) = 0 2 f ( x ) ( x 1)( x 4) = 0 x+1=0 OR x-4=0 x=-1 OR x=4 If f is a polynomial function and r is a real number for which f (r) = 0, then r is called a (real) zero of f, or root of f. If r is a (real) zero of f, then a.) (r,0) is an x-intercept of the graph of f. b.) (x - r) is a factor of f. x r m m 1 is a factor of a polynomial f and x r If is not a factor of f, then r is called a zero of multiplicity m of f. Example: Find all real zeros of the following function and their multiplicity. 5 1 2 f ( x ) x 3 x 7 x 2 x = 3 is a zero with multiplicity 2. x = - 7 is a zero with multiplicity 1. x = 1/2 is a zero with multiplicity 5. If r is a Zero of Even Multiplicity Sign of f (x) does not change from one side to the other side of r. Graph touches x-axis at r. If r is a Zero of Odd Multiplicity Sign of f (x) changes from one side to the other side of r. Graph crosses x-axis at r. Theorem: If f is a polynomial function of degree n, then f has at most n - 1 turning points. Theorem: For large values of x, either positive or negative, the graph of the polynomial f ( x ) a n x a n 1 x n n 1 a 1 x a 0 resembles the graph of the power function f ( x) a n x n For the polynomial 2 f ( x ) x 1 x 5 x 4 (a) Find the x- and y-intercepts of the graph of f. The x intercepts (zeros) are (-1, 0), (5,0), and (-4,0) To find the y - intercept, evaluate f(0) f (0) (0 1)(0 5)(0 4) 20 So, the y-intercept is (0,-20) For the polynomial 2 f ( x ) x 1 x 5 x 4 b.) Determine whether the graph crosses or touches the x-axis at each x-intercept. x = -4 is a zero of multiplicity 1 (crosses the x-axis) x = -1 is a zero of multiplicity 2 (touches the x-axis) x = 5 is a zero of multiplicity 1 (crosses the x-axis) c.) Find the power function that the graph of f resembles for large values of x. f (x) x 4 For the polynomial 2 f ( x ) x 1 x 5 x 4 d.) Determine the maximum number of turning points on the graph of f. At most 3 turning points. e.) Use the x-intercepts and test numbers to find the intervals on which the graph of f is above the x-axis and the intervals on which the graph is below the x-axis. On the interval x 4 Test number: x = -5 f (-5) = 160 Graph of f: Above x-axis Point on graph: (-5, 160) For the polynomial 2 f ( x ) x 1 x 5 x 4 On the interval 4 x 1 Test number: x = -2 f (-2) = -14 Graph of f: Below x-axis Point on graph: (-2, -14) On the interval 1 x 5 Test number: f (0) = -20 Graph of f: x= 0 Below x-axis Point on graph: (0, -20) For the polynomial 2 f ( x ) x 1 x 5 x 4 On the interval 5 x Test number: x=6 f (6) = 490 Graph of f: Above x-axis Point on graph: (6, 490) f.) Put all the information together, and connect the points with a smooth, continuous curve to obtain the graph of f. 500 (6, 490) 300 (-1, 0) (-5, 160) 100 8 (-4, 0) 6 (0, -20) 4 100 2 0 2 (-2, -14) 300 4 (5, 0) 6 8