Zeros of Polynomial Functions

Notes 3-5: Zeros of Polynomial Functions Name: ANSWER KEY Date: Period: Model & Discuss Charlie and Aisha built a small rocket and launched it from their backyard. The rocket fell to the ground 10 s after it launched. The height h, in feet, of the rocket relative to the ground at time t seconds can be modeled by the function shown. h(t) = a * t ^ 2 + bt + c A. How are launch and landing times related to the modeling function? The height of the rocket is O of both the launch and Parding. So O sceands and ID seerrids as the rules of t that sulm the equiding Oat, bl.c B. What additional information about the rocket launch could you use to construct an accurate model for the rocket's height relative to the ground? You nood informatia about the huget of the sound of evithe thee or Kowledge rickets shed spore and not guatatand erfoctor. Example 1: Use Zeros to Graph a Polynomial Function What are the zeros of f(x)= x(x - 4)(x + 3) ? Graph the function. 3 2 A duet P(x)=x+5x+6x+9 and D(x)=x+3. Aterm of a polynomad fuiting is a vale for what the forelion equils d. By the respondwet Property, the com a 8, 4, 3 This divide the y 20 10 X Interval Sign x - 4 x + 3 Product To see how the graph of the fonte benars to send interas, lord at the Sign of cand faster in card car and the sign of the pirant -8-40 ¥10 48 x < - 3 20 - 3 < x < 0 + 0 < x < 4 + x > 4 + + + + t + Skotid the graph. Draw a costinious curve that passes thorgt each Zere in the X AXIS and is below the X-ENS Nom the function 13 negate, and abve the xaxe won the fronten is positie Try It! Factor the function. Then use the zeros to sketch the graph. f(x) = 4x ^ 3 + 4x ^ 2 - 24x 0 = 4x(x ^ 2 + x - 6) 0=4x(x+ overline 2 ) (x - x) x = 0, - 3 20 y 10 A X 8-40 48 x < - 3 x ^ - 1 | 10 + + - 3 < x < 0 -20 0 < underline x( lambda + + + +++ x > 2 g(x) = x ^ 4 - 81 = (x ^ 2 - 9)(x ^ 2 + 4) = (x - 3)(x + 3)(x ^ 2 + 9) chi = 3, - 3 20 A 40 X (x-3) | (x + 3) x ^ 2 + z Pr + + -8-4 0 4 8 x < - 3 - 5 < x < 3 + + -40 x > 3 + + T + -20- f(x) = x ^ 3 - 3x ^ 2 + x - 3 x ^ 2 * (x - 3) + 1(x - 3) (x - 3)(x ^ 2 + 1) (x-3 (x ^ x + 1) g ma + x + 3 x < i x * 3 2. Turning Prinks + + + Example 2: Understand How a Multiple Zero Can Affect a Graph I low doca a multiple zoro affect the graph of a polynomial function? a zero whin faster appears meie than once in a polynomial graphs behave similar to the expirent when it is close to the zero. Ex. multiplicity 2 quadrethe multiplicly cutic - Eum multipleches torch and turn (x-axis) (turney print on the avis) Bad meltipliciter pass though Coveries the x-axis) Try It! Describe the behavior of the graph of the function at each of its zeros. f(x) = x(x+4)(x-1) Zerus: 0, -4,1 pars thugh the zon 1. The girph crusiss the the xaxis at the sons and D, and has a frining pous at f(x) = (x²+9)(x-1)(x + 2)² X-1-0 X+2=0 X-2 The graph crosses the x-axis at 1, 1, and has a trining port at the aere -2. Example 3: Find Real and Complex Zeros What are all the real and complex zeros of the polynomial function shown in the graph? Lising the graph, x = 2 is a zew. =814-6-6 D f(x) = 2 ^ 3 + 2 ^ 2 - 3(2) * 6 \mathcal{C}_{0}*finm : B_{3} the Factor Theoren x - 2 is a factor 201 10 X -4 -10 02 4 Find the other factus with Synthetic Division -20 aligned 1&1&-3&6\\ underline + 2 &6&6\\ 1&3&0 matrix f(x) = (x - 2)(x ^ 2 + 3x + 3) Use af to find other Zem. b = 3 f(x) = x ^ 3 + x ^ 2 - 3x - 6 x=- b pm sqrt 8Recx R^ a- 2a x = (- 3 plus/minus sqrt(3 ^ 2 - 4(1)(3)))/(3(1)) x = (- 3 plus/minus sqrt(9 + 12))/2 x = (- 3 plus/minus sqrt(- 3))/2 x = (- 3 plus/minus i * sqrt(3))/2 The zoros ar - 3/2 - (sqrt(3))/2 * i 2, - - 3/2 + (sqrt(3))/2 * i Try It! What are all the real and complex zeros of the polynomial function shown in the graph? f(x) = 2x ^ 3 - 8x ^ 2 + 9x - 9 y 5 X (x - 3) is a factor -4-20 2 4 4 4 8 - 4 * 4 2 -2 3 0 f(x) = (x - 3)(2x ^ 2 - 2x + 3) -5 -10 62 23 a = 2 x = - (b plus/minus sqrt(b ^ 2 - 4ac))/(2a) x = (2 plus/minus 2 * i * sqrt(5))/4 15 x = (2 plus/minus sqrt(4 - 4(2)(3)))/4 x = (1 plus/minus i * sqrt(5))/92 The Zeros pres chi = (2 plus/minus sqrt(4 * 24))/4 x = (2 plus/minus sqrt(2) * x)/1 1/2 - (sqrt(5))/2 * i 1/2 + (sqrt(5))/2 * i