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Real Zeros of Polynomial Functions Lesson 4.4

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Division of Polynomials Can be done manually See Example 2, pg 253 Calculator can also do division Use propFrac( ) function

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Division Algorithm For any polynomial f(x) with degree n ≥ 0 There exists a unique polynomial q(x) and a number r Such that f(x) = (x – k) q(x) + r The degree of q(x) is one less than the degree of f(x) The number r is called the remainder

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Remainder Theorem If a polynomial f(x) is divided by x – k The remainder is f(k)

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Factor Theorem When a polynomial division results in a zero remainder The divisor is a factor f(x) = (x – k) q(x) + 0 This would mean that f(k) = 0 That is … k is a zero of the function

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Completely Factored Form When a polynomial is completely factored, we know all the roots

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Zeros of Odd Multiplicity Given Zeros of -1 and 3 have odd multiplicity The graph of f(x) crosses the x-axis

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Zeros of Even Multiplicity Given Zeros of -1 and 3 have even multiplicity The graph of f(x) intersects but does not cross the x-axis

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Try It Out Consider the following functions Predict which will have zeros where The graph intersects only The graph crosses

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From Graph to Formula If you are given the graph of a polynomial, can the formula be determined? Given the graph below: What are the zeros? What is a possible set of factors? Note the double zero

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From Graph to Formula Try graphing the results... does this give the graph seen above (if y tic-marks are in units of 5 and the window is -30 < y < 30) The graph of f(x) = (x - 3) 2 (x+ 5) will not go through the point (-3,7.2) We must determine the coefficient that is the vertical stretch/compression factor... f(x) = k * (x - 3)2(x + 5)... How?? Use the known point (-3, 7.2) 7.2 = f(-3) Solve for k

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Assignment Lesson 4.4 Page 296 Exercises 1 – 53 EOO 73 – 93 EOO

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