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Polynomial Functions Let’s say I asked you to find all the roots of 10x3+9x2–19x+6=0. How can we solve this? 1. _______ 2. _______ 3. _______ This function cannot be factored using any method we’ve previously learned. We do know, though, that if a number is a zero of the function, it will produce a remainder of zero if synthetic division is used. But what number do we try? There are infinitely many numbers out there! Well, here lies the need for the… Rational Root Theorem: Let P be a polynomial function with integer coefficients in standard form. If p/q (in lowest terms) is a root of P(x) = 0, then p is a factor of the ______________ of P and q is a factor of the _________________ of P. p - factors of constant term 6: ___________________ q – factors of leading coefficient 10: _________________ p/q: ______________ Wow! That’s a lot of choices (although smaller than the previous list of infinitely many numbers). We would have to choose one, do _____________ and cross our fingers that we have a remainder of ________. If we do, we are good to go. If we don’t we have to pick another one and try again. Anyway, we DO have graphing calculators, so let’s use them! Ex 1. Let f(x) = 10x3 + 9x2 – 19x + 6 = 0 a) State the degree. b) Classify by the number of terms c) State the leading coefficient d) Find the rational roots Ex 2. Let f(x) = – 17x2 + 59x + 3x3 – 65 = 0 a) State the degree. b) Classify by the number of terms c) State the leading coefficient d) Find the rational roots You Try: Let f(x) = – 11x + 8x3 + 10x2 + 2 = 0 a) State the degree. b) Classify by the number of terms c) State the leading coefficient d) Find the rational roots For real numbers, these things mean the same thing: “solve the equation”, “find the zeros”, “find the x-intercepts”, “find the roots” Ex 4: Let f(x) = x2 + x +3x4 – x3 – 1 a) State the degree. b) Classify by the number of terms c) State the leading coefficient d) Is –2 a zero of the function? You Try: Let f(x) = x2 - 4x + x3 – 4 a) State the degree. b) Classify by the number of terms c) State the leading coefficient d) Is 2 a root of the function? Since the x-axis only represents ____ numbers, imaginary roots cannot be determined by using a ______. The graph of a polynomial function with ___ degree must cross the x-axis at least _____ (think end behavior!). The graph of a function with_____ degree __________ cross the x-axis. If it does, it will cross an even number of times. Ex 5: Sketch a graph with degree 3 that only has one real root. Ex 6: Sketch a graph with degree 3 that only has three real roots. You Try: Sketch a graph with degree 3 that only has two real roots. Every polynomial function of degree n > 1 has exactly n ________ zeros, counting multiplicities. If ____ is a root of a polynomial function with real-number coefficients, then _________ is also. You will always have an _____ number of ________ zeros. Ex 7: a) Write a polynomial equation of least degree with roots 2, 3i and –3i. b) Does the equation have an odd or even degree? c) How many times does the graph of the related function cross the x-axis? You Try: a) Write a polynomial equation of least degree with roots 4, 3i and –3i. b) Does the equation have an odd or even degree? c) How many times does the graph of the related function cross the x-axis? Ex 8: Write a polynomial function, P, with the given information: P(0) = 120; zeros of p(x) are -3, 2, 4 You Try: Write a polynomial function, P, with the given information: P(0) = 100; zeros of p(x) are 5, -2, 5 Ex 9: Write a polynomial function, P, with the given information: P(0) = 120; zeros of p(x) are -3, 2, 4, 5i, -5i You Try: Write a polynomial function, P, with the given information: P(0) = 100; zeros of p(x) are -6i, -2, 5 Ex 10: Use the equation 32x3 – 32x2 + 4x – 4 = 0. a) State the number of complex roots. b) Factor the polynomial by grouping. c) Find the roots. You Try: Use the equation 4x3 – 2x2 + 16x – 8 = 0. a) State the number of complex roots. b) Factor the polynomial by grouping. c) Find the roots. Ex 11: When a gold ball is hit from a tee with a velocity of 160 ft/s at an angle of 45o with respect to the ground, the height(in feet) of the ball above the ground is given by h(x) = x2 , where x is the horizontal distance from the tee. x 800 a) How far from the tee does the ball strike the ground? b) Verify your answer using a graph. You Try: A meteorologist sends a temperature probe on a small weather rocket through a cloud layer. The launch pad for the rocket is 2 feet off the ground. The height of the rocket after launching is modeled by the equation h = -16t2+232t+2, where t is the elapsed time in seconds. a) How far will the rocket be after 14 seconds?? b) Verify your answer using a graph.