Chapter 7 Polynomial Functions Factoring Review We will talk about 3 Types: 1. 2. 3. ALWAYS FACTOR OUT FIRST! Ex 1: Factor x 2 + 5x + 6 Ex 2: Factor x 2 + 16x + 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
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 + 120 = 30 + 11x
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 2 + 59x + 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.
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:
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?
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 3 + 5 by x 2 + 2
Ex 4: Find (10x 3 + 32x 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
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 2 + 24x b. What are the zeros for the graph of p(x) Ex 2: a. Factor p(x) = 5x 3 + 11x 2 + 24 b. What are the zeros for the graph of p(x)
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 -624. 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.
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. 2 17 Complex Numbers
Ex 2: Solve x 2 2x + 7 = 0 Ex 3: Solve 3z 2 + 10 = 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
Ex 6: Express 4 3i in standard form 5+i Ex 7: Factor x 2 + 9 Ex 8: Factor 4x 2 + 81 7.6 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
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 2 + 144x 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)?
Conjugate Zeros Theorem Ex 4: Let g(x) = 3x 3 2x 2 + 12x 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)
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 3 + 125 Ex 3: Factor 27a 3 8b 6 Ex 4: Factor x 6 64 Sum/Difference4 of Odd Powers
Ex 5: Factor a 7 + b 7 Ex 6: a. Factor g(x) = x 3 + 27 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
Ex 10: Find all real solutions to x 6 = 1 by factoring Ex 11: Find all real solutions to x 4 = 289 by factoring