Graph Properties of Polynomial Functions
Ch.2 Polynomial FunctionsLesson 2.1 PolynomialsObjective: Students will be able to identify a polynomial function, to evaluate it using synthetic substitution and to determine its zeros. Polynomial: can be written in the form…..Standard form: when the powers of x are written in descending order. Leading term:leading coefficient:degree: NameDegreeExampleConstantLinearQuadraticCubicQuarticQuinticExample: State whether the following are polynomial functions. Give the zeros of the function, if they exist. F(x) = 2x3 – 32xb. fx= x+1x-1Example: If P(x) = 3x4-7x3-5x2+9x+10, evaluate the following.P(2)b. P(-3n)Synthetic Substitution: another way to evaluate polynomials Example: P(x) = 3x4-7x3 – 5x2 + 9x + 10, find P(2) S(x) = 3x4-5x2+9x+1-, find S(-2)Synthetic Division: Used to divide a polynomial by a binomial of the form x-aDivide 2x4 – 15x2 -10x + 5 by x-3Examples: Use synthetic division to divide the polynomial by the linear factor. 1. (3x2+ 7x + 2) ÷ (x + 2) 2. (2x2+ 7x – 15) ÷ (x + 5)3. (7x2– 3x + 5) ÷ (x + 1) 4. (4x2+ x + 1) ÷ (x – 2) 5. (3x2+ 4x – x4– 2x3– 4) ÷ (x + 2) 6. (3x2– 4 + x3) ÷ (x – 1)HW: p. #2-10 even, 16, 19, 20Lesson 2.2 Synthetic Division, The Remainder and Factor TheoremsObjective: Students will be able to use synthetic division and to apply the remainder and factor theorems. DO NOW: Divide the following using synthetic division. X3 – 2x2 + 5x + 1; x-1b. x4 – 2x3 + 5x + 2; x-1Use Synthetic Substitution to find P(3) if P(x) = 3x3 – 4x2 + 2x -5THE REMAINDER THEOREM: When a polynomial P(x) is divided by x-a, the remainder is P(a). Ex. Find the remainder when P(x) is divided by the given divisor. P(x) = 5x3 – 4x2 – 2x + 1P(1) b. P(4) THE FACTOR THEOREM: For a polynomial P(x), x-a is a factor if and only if P(a) = 0. Ex. If P(x) = 2x4 + 5x3 – 8x2 = 17x -6, determine whether each of the following is a factor of P(x):x-1b. x-2 Ex. Is x+1 a factor of P(x) = x3 + 3x2 + x -1? If so, find the other factors of P(x). Ex. You are given a polynomial equation and one or more of its roots. Find the remaining roots. 2x3 – 5x2 – 4x + 3; root: x=3b. 2x4-9x3+2x2 + 9x – 4; roots: x=-1, x=1HW: p. 61 # 12-24 even Graphs of Polynomial FunctionsObjective: Students will be able to describe end behavior using limit notation, find the zeros of a function and use multiplicity rules to graph. DO NOW: Graph the following parent functions. Make a table or use your graphing calculator 020955000320040026670000y = xy=x25715030607000y=x3 y = x45245108191500Graph Properties of Polynomial FunctionsLet P be any nth degree polynomial function with real coefficients.The graph of P has the following properties.P is continuous for all real numbers, so there are no breaks, holes, jumps in the graph.The graph of P is a smooth curve with rounded corners and no sharp corners.3. The graph of P has at most n x-intercepts.4. The graph of P has at most n – 1 turning points.End behavior of polynomials: What happens to the graph as x approaches ±∞*The end behavior of a polynomial is closely related to the end behavior of its leading term. *Example: Comparing the Graphs of a Polynomial and its Leading TermGraph f(x) = x3 and g(x) = x3 - 4x2 - 5x - 3 in the calculator.Continue zooming out. What happens?In general….Leading Term Test for Polynomial End BehaviorFor any polynomial function, the limits as x goes to positive and negative infinity are determined by the degree n of the polynomial and its leading coefficient An.An> 0 and n is oddAn< 0 and n is oddAn> 0 and n is evenAn<0 and n is evenExample 1: Given the following polynomial functions, state the degree of the function and describe the end behavior using limit notation. a. b. Example: Find the zeros of the polynomial and use the end behavior to draw a rough sketch of the graph. 228600018034000Even multiplicity: touches x-axis, but doesn’t cross (looks like a parabola there).Odd multiplicity of 1: crosses the x-axis (looks like a line there).Odd multiplicity >3: crosses the x-axis and looks like a cubic there.Example: Determine the end behavior, zeros, and multiplicity of each zero for the following. Then graph.P(x) = (x+2)3(x-1)2Lesson 2.4 Rational Root TheoremObjective: Students will be able to apply the rational root theorem to test roots in order to determine the zeros of a function. DO NOW: Find the roots of the following polynomials. f(x) = 15x3 + 5x2 + 3x + 1b. f(x) = 4n3 – 12n2 + 3n -9Let’s look at the calculator to determine the zeros of Now, let’s look at the calculator to determine the zeros of P(x) = 2x4?+?x3?– 31x2?– 26x?+ 24The Rational Zero Theorem:WHY IS IT IMPORTANT: Narrows the search for rational zeros to a finite list.If has integer coefficients and is a rational zero (in lowest terms) of p, then p is a factor of the constant term and q is a factor of the leading coefficient . Use the Rational Zeros Theorem to list all the Possible Rational Zeros of the following polynomialsEX1: f(x) = X3 – 4x2 + 15EX2: f (x) = 10x3 + 7x2-2x-6EX3: Find the roots of f(x) = x3 – 2x2 -11x +12HINT: Apply the Rational Root Theorem to find the possible rational roots!What is p? ________What is q? ________EX4: Find the roots of f(x) = 3x3 – x2 -15x + 5EX 5] Find the possible rational roots of . Lesson 2.5 Conjugate PairsObjective: Students will be able to find conjugate pairs of complex zeros. They will also be able to write a polynomial function given the zeros. Do now: Find the polynomial with integer coefficients having roots at?3, –5, and?–?,? and passing through?(–1, 16). Write the answer in standard form. Find the zeros of the given polynomial. P(x) = x4 – 3x3 + 6x2 + 2x – 60What do you notice about the complex zeros in example b?Complex Conjugate Zeros Theorem:If f is a polynomial function and a +bi is a zero of f , then a - bi is also a zero of f .Irrational Conjugate Zeros Theorem:If f is a polynomial function and 2+3 is a zero of f , then 2-3 is also a zero of f .EX: Find a 3rd degree polynomial in standard form with integer coefficients given -2, and 3i are zeros. EX: Find a polynomial with integer coefficients given the zeros at 1 and 2 - 3i.EX: Write a polynomial with integer coefficients with degree 4 and zeros at -2 (multiplicity 2) and - i .EX: Write a polynomial with integer coefficients with zeros at 4 and 3 + 2.EX: Use the given zero to find the remaining zeros of each polynomial function. P(x) = 2x3 – 5x2+6x -2 Zero: 1+iHW: Worksheet ................
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