Real Zeros of Polynomial Functions

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Real Zeros of Polynomial Functions Lesson 4.4. Division of Polynomials Can be done manually See Example 2, pg 253 Calculator can also do division Use.

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Presentation transcript:

Real Zeros of Polynomial Functions Lesson 4.3

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

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

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

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

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

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

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

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

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

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

Assignment Lesson 4.3A Page 265 Exercises 1 – 45 odd Lesson 4.3B Exercises 47 – 57 and 91 – 97 odd