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Polynomial Functions Unit 4 Polynomials • Monomial—a number, variable, or product of numbers and variables all raised to whole number powers • Polynomial Expression--a monomial or sum of monomials. • Polynomial Function—function that is defined by a polynomial expression • Leading coefficient—coefficient of highest powered term • Degree of polynomial—power of highest powered variable Standard Form The standard form of a polynomial function arranges the terms by degree in descending numerical order A polynomial function P(x) in standard form is: 𝑃 𝑥 = 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 +. . . +𝑎1 𝑥 + 𝑎0 Where n is a nonnegative integer and 𝑎𝑛 , . . . 𝑎0 are real numbers. Ex: 𝑃 𝑥 = 4𝑥 3 + 3𝑥 2 + 5𝑥 − 2 Classifying Polynomials by degree • d=0 Constant 52 • d=1 Linear x2 • d=2 Quadratic x 2x 1 • d=3 Cubic 4x 6 • d=4 Quartic 9 x 8x 2 x x 2 • d=5 Quintic x 3x 2 x 2 3 4 5 3 4 2 Classifying by Number of Terms • Monomial—one term • Binomial—two terms • Trinomial—three terms • N-nomial—n terms Classify by degree and number of terms. 1. 3𝑥 2 + 2𝑥 − 3 2. 4𝑥 5 − 3𝑥 3. 2𝑥 2 − 4𝑥 4 + 6𝑥 4. 4 𝑥 7 Polynomial Function • Polynomial equation used to represent a function f ( x) 4 x 2 5 x 2 P( x) 10 x 3 2 x 2 x 3 Graphs of Polynomial Functions • Constant Linear Quadratic • Show graphs with positive and negative LC End Behavior of Graphs As 𝑥 → ∞, 𝑓(𝑥) → _________ As 𝑥 → −∞, 𝑓(𝑥) → _________ As x gets bigger or smaller, what happens to the function value? Graphs of Polynomial Functions • Quadratic: 𝑓 𝑥 = 𝑥 2 − 2𝑥 D:________ R: ________ Zeros: _______ Inc: _______ Dec: _______ as 𝑥 → ∞, 𝑓(𝑥) → _________ as 𝑥 → −∞, 𝑓(𝑥) → _________ Graphs of Polynomial Functions • Cubic: 𝑓 𝑥 = 𝑥 3 + 𝑥 2 D:________ R: ________ Zeros: _______ Inc: _______ Dec: _______ as 𝑥 → ∞, 𝑓(𝑥) → _________ as 𝑥 → −∞, 𝑓(𝑥) → _________ Graphs of Polynomial Functions • Quartic: 𝑓 𝑥 = 4𝑥 4 − 7𝑥 2 − 2 D:________ R: ________ Zeros: _______ Inc: _______ Dec: _______ as 𝑥 → ∞, 𝑓(𝑥) → _________ as 𝑥 → −∞, 𝑓(𝑥) → _________ Graphs of Polynomial Functions • Quintic: 𝑓 𝑥 = 𝑥 5 − 4𝑥 3 D:________ R: ________ Zeros: _______ Inc: _______ Dec: _______ as 𝑥 → ∞, 𝑓(𝑥) → _________ as 𝑥 → −∞, 𝑓(𝑥) → _________ End Behavior of Graphs +Lead Coefficient Even Degree Odd Degree -Lead Coefficient End Behavior of Graphs As 𝑥 → ∞, 𝑓(𝑥) → _________ As 𝑥 → −∞, 𝑓(𝑥) → _________ End Behavior of Graphs As 𝑥 → ∞, 𝑓(𝑥) → _________ As 𝑥 → −∞, 𝑓(𝑥) → _________ To sketch the graph • Determine the end behavior. • Determine the x intercepts (where y=0) f ( x) x 2 2 x 0 x2 2x 0 x( x 2) x 0 and x 2 0 x 0 and x 2 Turning Points A polynomial function of degree n has at most n-1 turning points. 1. 𝑓 𝑥 = 3𝑥 5 − 4𝑥 3 + 3𝑥 2 + 2 2. 𝑓 𝑥 = 𝑥 3 − 3𝑥 4 + 2𝑥 2 − 1 3. 𝑦 = 𝑥 2 + 2𝑥 + 3