Section 4.1: Polynomial Functions and Models Learning Objectives: 1. Identify Polynomial Functions and Their Degree 2. Graph Polynomial Functions Using Transformations 3. Identify the Real Zeros of a Polynomial Function and Their Multiplicity 4. Analyze the Graph of a Polynomial Function 5. Build Cubic Models from Data Page 1 of 19
Function Factors Graph Zeros Multiplicity ( ) 2 f x = x 3x 10 ( ) 2 f x = x + 4x+ 4 3 f ( x) = x x ( ) 3 2 f x = x 6x + 9x ( ) 2 f x = 2x + 1 Page 2 of 19
n n f ( x) = ax, n is even f ( x) ax 1 2 f ( x) = x ( ) = 5 2 =, n is odd f x x 4 3 f ( x) = x f ( x) = x f x = x 2 1 5 f ( x) = 3x ( ) 1 6 3 f ( x) = x ( ) 2 2 f x = 2x 1. 2. 3. 4. 1. 2. 3. 4. Page 3 of 19
Find a polynomial f of degree 3 whose zeros are -4, -2, and 3. Page 4 of 19
Determine the polynomial function whose graph is given. Page 5 of 19
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Summary Analyzing the Graph of a Polynomial Function Step 1: Step 2: Step 3: Step 4: Step 5: Step 6: Step 7: Step 8: Page 7 of 19
1. Determine [algebraically] the quadratic function whose graph is given. Write in standard form: f(x) = ax 2 + bb + c, a 0You must show work to receive credit. (0,9) 2. Both graphs below are of the same polynomial function [in different viewing windows]. The one on the left is not a complete graph. 200 1500 100 0-7 -6-5 -4-3 -2-1 0 1 2 3 4 5 6-100 -200 500-7 -6-5 -4-3 -2-1 0 1 2 3 4 5 6-500 Write an equation of the graph pictured that includes the points (0, 0), (-5, 0), (4, 0), (-1, 0) and (1, 216). Leave your equation in factored form. Page 8 of 19
3. The following statements about the polynomial function f are true. (5 points) f has exactly three zeros. The graph of f has the following end behavior: as x, f(x) and as x, f (x). For each of the following statements circle T if the statement must always be true, F if the is always false, or S if the statement is sometimes false and sometimes true. T F S The leading coefficient is negative. T F S f(x) is an odd function. T F S The graph of f has at most three local extrema (local maxima/minima). T F S f(x) is a third degree polynomial function. T F S f(x) has at most two x-intercepts. 4. Page 9 of 19
Finding Zeros of Polynomial Functions (4.2 & 4.3) EXAMPLE: Using the Remainder Theorem Is (x 1) a factor of f(x) = 2x 3 + 11x 2 7x 6? Is (x + 4) a factor? Page 10 of 19
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Find the real zeros of the polynomial function f(x) and completely factor. 1. f(x) = 9x 3 + 7x 2 45x 35 2. f(x) = 7x 4 + 41x 3 + 78x 2 + 44x 8 3. f(x) = 7x 3 + 13x 2 9x + 1 Page 13 of 19
Find the real zeros of f rounded to decimal places: Page 14 of 19
Section 4.3 Complex Zeros; Fundamental Theorem of Algebra Find the zeros of the polynomial f(x) = x 2 + 1 and factor. Page 15 of 19
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Find a Polynomial Function with Specified Zeros Find a polynomial function f of degree 4 whose coefficients are real numbers and that has zeros specified zeros: 1, 1, and 4 + i Hint: If r = a + bb and r = a bb, then when the linear factor (x r) and (x r ) are multiplied, we have (x r)(x r ) = x 2 (r + r )x + rr Page 17 of 19
Find the complex zeros of f(x) = 7x 3 16x 2 + 74x 20 and completely factor. Page 18 of 19
Find the remaining zeros of h(x) = x 4 4x 3 + 32x 13; zero: 3 2i Use synthetic or long division. Page 19 of 19
Section 4.6 Fundamental Theorem of Algebra Every complex polynomial function f (x) of degree n > 1 has at least one complex zero. Theorem Every complex polynomial function f (x) of degree n > 1 can be factored into n linear factors (not necessarily distinct) of the form Conjugate Pairs Theorem Let f (x) be a complex polynomial whose coefficients are real numbers. If r = a + bi is a zero of f, then the complex conjugate is also a zero of f. Corollary A complex polynomial f of odd degree with real coefficients has at least one real zero.