Chapter 7 Polynomial Functions. Factoring Review. We will talk about 3 Types: ALWAYS FACTOR OUT FIRST! Ex 2: Factor x x + 64
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1 Chapter 7 Polynomial Functions Factoring Review We will talk about 3 Types: ALWAYS FACTOR OUT FIRST! Ex 1: Factor x 2 + 5x + 6 Ex 2: Factor x x + 64 Ex 3: Factor 4x 2 + 6x 18 Ex 4: Factor 4x 2 15x 25 Ex 5: Factor x 2 9 Ex 6: Factor 3x 2 48
2 Ex 7: Factor 16x 2 25y 2 Ex 8: Factor x 3 7x 2 + 2x 14 Ex 9: Factor 3x 2 + xy 12x 4y Ex 10: Solve by factoring 7x 2 6x = 0 Ex 11: Solve by factoring x 2 11x + 19 = 5 Ex 12: Solve by factoring 5x 2 44x = x
3 7.1 Characteristics of Polynomial Functions A polynomial in x is an expression of the form Where n is a nonnegative integer and a n 0. Degree o Ex: Coefficients o Ex: Leading coefficient o Ex: Standard form of a polynomial o Ex: Ex 1: Let p be the polynomial function with equation p(x) = 8x x + 21 a. What is the degree of p? b. What is the leading coefficient of p? c. Find the y intercepts of its graph. d. Find all x-intercepts of its graph.
4 Ex 2: Let f be a polynomial function with equation f(x) = 2(x 1) 3 (x + 4) 2 a. What is the degree of p? b. What is the leading coefficient of p? c. Find the y intercepts of its graph. d. Find all x-intercepts of its graph. Extrema of Functions 1. Maximum 2. Minimum Two Types of Extrema 1. Relative a. b. 2. Absolute a. b. Examples:
5 Intervals of Increasing/Decreasing Ex 3: Consider the graph of f(x) = x 3 5x 2. A relative maximum occurs when x 1.3 and a relative minimum occurs when x 1.3. Describe the intervals on which a. f is increasing b. f is decreasing Ex 4: A company wants to produce an open-top box from 60-cm by 45-cm piece of cardboard. They want the box to have the largest possible volume. The picture shows the way the cardboard will be cut. a. Write an equation for the volume V of the box in terms of x. b. What are the values of x for which this box would have zero volume? c. Graph the function in a graphing utility d. For what values of x does V have meaning in this situation? e. What is the relative maximum on this graph? f. For what values is the graph increasing or decreasing?
6 7.3 Division and the Remainder Theorem Long Division Ex 1: Divide 6x 2 + x 2 by 3x + 2 Ex 2: Divide 27x 3 9x 2 3x 10 by 3x 2 Ex 3: Divide 3x 4 7x by x 2 + 2
7 Ex 4: Find (10x x 2 4x 11) (10x + 2) Synthetic Division Ex 5: Divide 3r 4 8r 3 + r + 20 by r + 2 Remainder Theorem If a polynomial p(x) is divided by, then the remainder is. *In other words, Ex 6: Use the remainder theorem to find f(5) given f(x) = 4x 5 x 3 + 1
8 7.4 The Factor Theorem How are Zeros of a Polynomial Related to the Factors of the Polynomial? The function f(x) = 2x 3 + 2x 2 34x + 30 is graphed to the right. Find the zeros of the function. Factor Theorem For a polynomial p(x), is a of p(x) iff. *In other words Ex 1: a. Factor p(x) = x 4 6x 3 4x x b. What are the zeros for the graph of p(x) Ex 2: a. Factor p(x) = 5x x b. What are the zeros for the graph of p(x)
9 Ex 3: Find an equation for the 3 rd degree polynomial function p graphed to the right. Ex 4: Find an equation for the 4 th degree polynomial function f graphed to the right. Ex 5: The x-intercepts of the graph of y = r(x), where r(x) is a polynomial of degree 3 are -8, 12 and 13. The y- intercept of the graph is Find an equation for r(x). Ex 6: Find a 3 rd -degree polynomial p(x) with integer coefficients whose zeros are 1, 3 5, 2 3.
10 7.5 Complex Numbers The graph at the right has no x-intercepts which means it has no REAL zeros. But its zeros are the solutions to: The number i Ex 1: Find the following square roots a. 16 b. 48 c. i 108 d Complex Numbers
11 Ex 2: Solve x 2 2x + 7 = 0 Ex 3: Solve 3z = 5z Ex 4: Complete the following operations with complex numbers a. (3 + 2i) + ( 6 + 3i) b. ( 1 2i) ( 3 + 7i) c. (5 + 4i)( 2 8i) Complex Conjugates Ex 5: Find the complex conjugate of 3 + 2i
12 Ex 6: Express 4 3i in standard form 5+i Ex 7: Factor x Ex 8: Factor 4x The Fundamental Theorem of Algebra Function f(x) = x 2 6x + 8 g(x) = x 2 6x + 9 h(x) = x 2 6x + 10 Graph Discriminant Zeros Factors
13 Multiplicity When the discriminant = 0, the quadratic function has one real zero but the factor appears twice. Ex: Thus, every quadratic polynomial with real coefficients has EXACTLY complex zeros Fundamental Theorem of Algebra If p(x) is any polynomial of degree n 1 with complex coefficients, then p(x) has at least on zero. A polynomial of degree n 1 with complex coefficients has exactly complex zeros, if are counted Ex 1: Find the zeros of p(x) = x 3 24x x and indicate their multiplicities Ex 2: Find all zeros and their multiplicities of f(t) = 9t 15 6t 14 + t 13 A polynomial of degree can have at most turning points. Ex 3: A polynomial equation y = f(x) is graphed below. What is the lowest possible degree of f(x)?
14 Conjugate Zeros Theorem Ex 4: Let g(x) = 3x 3 2x x 8 a. Verify that 2i is a zero of g(x) b. Find the remaining zeros of g(x) Ex 5: Let f(x) = 4x 3 + 6x 2 + 5x + 3 a. Verify that 1 is a zero of f(x) b. Find the remaining zeros of f(x)
15 7.7/7.8 Factoring Sums and Difference of Powers and Advanced Factoring Techniques Sum/Difference of Cubes Ex 1: Factor x 3 64 Ex 2: Factor x Ex 3: Factor 27a 3 8b 6 Ex 4: Factor x 6 64 Sum/Difference4 of Odd Powers
16 Ex 5: Factor a 7 + b 7 Ex 6: a. Factor g(x) = x b. Find all the zeros of the polynomial Ex 7: Factor t 3 3t 2 4t + 12 Ex 8: Factor (x 2 + 4x) 2 + 7(x 2 + 4x) + 12 Ex 9: Factor x 2 + 2xy + y 2 + x + y 2
17 Ex 10: Find all real solutions to x 6 = 1 by factoring Ex 11: Find all real solutions to x 4 = 289 by factoring
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