Sec 3.1 Polynomial Functions Fundamental Theorem of Algebra An important and famous German mathematician, Carl Friedrich Gauss, is credited with first proving the FUNDAMENTAL THEOREM OF ALGEBRA which states: Every polynomial equation of degree 1 or greater has at least one root in the set of complex numbers. We can use an extension of this theorem to suggest that any polynomial of degree n, must have n complex linear factors. 1. How many complex linear factors must each of the following polynomials have? a. x 4 3x 3 + 4x 2 7x + 1 b. 2x 3 + 3x 5 5x 6 2x + 6 Name: Number of Complex Linear Factors : Number of Complex Linear Factors : 2. Consider the polynomial function f(x) is shown in the graph. Answer the following questions. a. List all of the zeros of f(x). b. Assuming all of the factors of the polynomial are real and the leading coefficient is 1, create a polynomial function in factored form that should describe f(x). f(x) = c. Rewrite the polynomial function, f(x), in expanded form. f(x) = (Compare the degree, number of linear factors, and number of zeros.) 3. Consider the polynomial function h(x) is shown in the graph. Answer the following questions. a. Create a polynomial function in factored form for h(x),using the graph and given that h(x) has complex zeros at x = i and x = i. h(x) = b. Rewrite the polynomial function, h(x), in expanded form. h(x) = (Compare the degree, number of linear factors, and number of zeros.) M. Winking Unit 3-1 page 42
4. Consider the polynomial function g(x) that has zeros at x = 3, x = 2, and x = 2 a. What is the minimum degree of the polynomial function g(x). b. Assuming all of the coefficients of the polynomial are real and the leading coefficient is 2, create the polynomial function in factored form that should describe g(x). g(x) = c. Rewrite the polynomial function, g(x), in expanded form. g(x) = 5. Consider the polynomial function p(x) that has zeros at x = 2, x = 2, and x = 4 a. What is the minimum degree of the polynomial function p(x). b. Assuming all of the coefficients of the polynomial are real and the function passes through the point (1, 27), create an algebraic polynomial in factored form that should describe p(x). p(x) = c. Rewrite the polynomial function, p(x), in expanded form. p(x) = 6. Consider the polynomial function q(x) that has zeros at x = 1 and x = 3i, a. What is the minimum degree of the polynomial function q(x). b. Assuming all of the coefficients of the polynomial are real and the leading coefficient is 1, create the polynomial function in factored form that should describe q(x). q(x) = c. Rewrite the polynomial function, q(x), in expanded form. q(x) = M. Winking Unit 3-1 page 43
7. Consider the polynomial function m(x) is shown in the graph that has a zero of multiplicity 2. Answer the following questions. a. List all of the zeros of m(x) and note any zeros that have a multiplicity of 2 or higher. b. Assuming all of the factors of the polynomial are real and the leading coefficient is 1, create a polynomial function in factored form that should describe m(x). m(x) = c. Rewrite the polynomial function, m(x), in expanded form. m(x) = (Compare the degree, number of linear factors, and number of zeros.) 8. Based on the degree of the polynomial function and the graph determine how many real and how many imaginary zeros the polynomial must have. Also identify any zero s having a higher multiplicity. a. f(x) = x 3 4x 2 + x + 6 b. h(x) = x 4 + 5x 2 + 1 c. g(x) = x 5 + 2x 4 4x 2 5x + 6 Number of Real Zeros: (Indicate any with higher multiplicity.) Number of Real Zeros: (Indicate any with higher multiplicity.) Number of Real Zeros: (Indicate any with higher multiplicity.) Number of Imaginary Zeros: Number of Imaginary Zeros: Number of Imaginary Zeros: M. Winking Unit 3-1 page 44
Common Polynomial Identities: Sec 3.2 Polynomial Functions Factoring Using Polynomial Identities Name: Description Identity Example Difference of Two Squares a 2 b 2 = (a + b)(a b) 9x 2 4y 2 = (3x + 2y)(3x 2y) Sum of Two Squares a 2 + b 2 = (a + bi)(a bi) 16m 2 + 9 = (4m + 3i)(4m 3i) Perfect Square Trinomial a 2 + 2ab + b 2 = (a + b) 2 9a 2 + 24a + 16 = (3a + 4) 2 Perfect Square Trinomial a 2 2ab + b 2 = (a b) 2 25p 2 30pq + 9q 2 = (5p + 3q) 2 Binomial Cubed a 3 + 3a 2 b + 3ab 2 + b 3 = (a + b) 3 x 3 + 6x 2 + 12x + 8 = (x + 2) 3 Binomial Cubed a 3 3a 2 b + 3ab 2 b 3 = (a b) 3 b 3 9b 2 + 27b 27 = (b 3) 3 Difference of Two Cubes a 3 b 3 = (a b)(a 2 + ab + b 2 ) 8w 3 27 = (2w 3)(4w 2 + 6w + 9) Sum of Two Cubes a 3 + b 3 = (a + b)(a 2 ab + b 2 ) 64y 3 + 1 = (4y + 1)(16y 2 4y + 1) 1. Factor the following using a Difference or Sum of Two Squares. a. 4a 2 25b 2 b. (5m 3 ) 2 (6n) 2 c. a 2 b 8 9p 6 q 2 d. x 2 + 36 e. 18p 2 98q 2 f. 16w 2 + 7 M. Winking Unit 3-2 page 45
2. Factor the following using a Difference or Sum of Two Cubes. a. a 3 64 b. 27x 3 + 8y 6 c. 16m 9 250 d. (7p 2 ) 3 + (2q 4 ) 3 3. Verify the following polynomial identity from each side. a. (a 2 + b 2 ) 2 = (a 2 b 2 ) 2 + (2ab) 2 b. (a 2 + b 2 ) 2 = (a 2 b 2 ) 2 + (2ab) 2 c. This specific identity is commonly used to find sets of Pythagorean triples. i. Find the Pythagorean triple that would be created by using a = 3 and b = 2. ii. Find the Pythagorean triple that would be created by using a = 5 and b = 2. M. Winking Unit 3-2 page 46
4. Verify the following polynomial identity. x 2 + y 2 + z 2 + 2(xy + xz + yz) = (x + y + z) 2 M. Winking Unit 3-2 page 47
Factor the following polynomial expressions. Sec 3.3 Polynomial Functions Alternate Factoring Methods Name: 1. 7xy + 10y 5zy 2. 3x(2x 1) + 4(2x 1) + 2x(2x 1) 3. a 2 (a + 3) 9(a + 3) 4. (2a + 7)(3b 1) + (3a 2)(3b 1) Factor the following polynomial expressions using the GROUPING technique. 5. x 3 4x 2 + 5x 20 6. 6m 3 + 8m 2 + 9m + 12 7. 9a 3 6a 2 6a + 4 8. 3p 3 2p 2 27p + 18 M. Winking Unit 3-3 page 48
Factor the following polynomial functions by analyzing the Graph of the function to help you find linear factors based on the zeros of the function. 9. f(x) = x 3 2x 2 x + 2 10. g(x) = x 3 4x 2 + 4x 11. m(x) = x 3 x 2 6x + 6 12. q(x) = x 4 x 3 5x 2 + 3x + 2 13. h(x) = x 3 + 4x 2 + 4x + 16 14. w(x) = x 4 +x 3 + 2x 2 6x + 4 M. Winking Unit 3-3 page 49
Sec 3.4 Polynomial Functions Rational Root Theorem & Remainder Theorem Rene Descartes is commonly credited for devising the Rational Root Theorem. The theorem states: Given a polynomial equation of the form Name: 0 = a 0 x n + a 1 x n 1 + a 2 x n 2 + + a n 1 x + a n Any rational root of the polynomial equation must be some integer factor of a n divided by some integer factor of a 0 Given the following polynomial equations, determine all of the POTENTIAL rational roots based on the Rational Root Theorem and then using a synthetic division to verify the most likely roots. 1. x 3 + x 2 8x 12 = 0 2. 4x 3 12x 2 + 5x + 6 = 0 Potential Rational Roots: Potential Rational Roots: The Remainder Theorem suggests that if a polynomial function P(x) is divided by a linear factor (x a) that the quotient will be a polynomial function, Q(x), with a possible constant remainder, r, which could be written out as: P(x) = (x a) Q(x) + r If this seems a little complicated consider a similar statement but just using integers. For example (using the same colors to represent similar parts), 70 6 = 11 with remainder 4 which could also be rewritten as: 70 = 6 11 + 4 The Remainder Theorem also leads to another important idea, The Factor Theorem. To state the Factor Theorem, we only need to evaluate P(a) from the Remainder Theorem. P(a) = (a a) Q(a) + r :Substitute a in for each x P(a) = (0) Q(a) + r :Simplify (a a) = 0 P(a) = r :Simplify 0 Q(a) = 0 This is an important fact that basically states the remainder of the statement P(x) (x a) is P(a). M. Winking Unit 3-4 page 50
Using the Remainder or Factor Theorem answer the following. 3. Using Synthetic Division evaluate x 3 + x 2 8x 12 when x = 3 4. Use Synthetic Division to find the remainder of (x 3 + x 2 8x 12) (x 3) 5. Using Synthetic Division evaluate f( 2) given f(x) = 3x 4 + 7x 2 8x + 12. 6. Use Synthetic Division to determine the remainder of the quotient of f(x) and g(x), given f(x) = 3x 4 + 7x 2 8x + 12 and g(x) = x + 2 7. Given f(x) = (x + 5) Q(x) + 8, evaluate f( 5). 8. Given f(x) = Q(x) with a remainder 3, (x 6) evaluate f(6). 9. Consider f(x) = 2x 3 1x 2 + 3x + 4 and that f(b) = 5 10. Consider g(x) = x 3 + 3x 2 2x + 4 and Justin used synthetic division to divide(x 3 + 3x 2 2x + 4) (x k). His work is partially shown below. Using this information determine f(k). What value should be in box labeled a? M. Winking Unit 3-4 page 51a
Using any available techniques determine the following (find exact answers). 11. Find all of the solutions to the polynomial equation x 4 3x 3 + 6x 2 12x + 8 = 0 12. Find all zeros of the polynomial function f(x) = x 4 4x 3 + x 2 + 8x 6 M. Winking Unit 3-4 page 51b
ANALYZING GRAPHS PROBLEM BUTTONS CALCULATOR Locate a zero of the Function: y x 4 3x 3 2x 1. First graph the function Y =, (-), X,T,,n, ^, 4,, 3, X,T,,n, ^, 3, +, 2, X,T,,n, ZOOM, 6 2. Press, 2 nd, TRACE, 2 3. Using the LEFT arrow key,, move the cursor to the left of the zero that is to be determined and press ENTER. 4. Using the Right arrow key,, move the cursor to the right of the zero that is to be determined and press ENTER. 5. Finally, using the arrow keys,,, move the cursor to the approximate location of the zero that is to be determined and press ENTER. ( 2.73, 1.4E 12) **NOTICE the ordinate (y-value) doesn t precisely compute to zero but 1.4E 12 means 1.4 12 10 which is extremely close to zero. Matt Winking mmwinking@gmail.com
Sec 3.5 Polynomial Functions Polynomial Symmetry 1. Describe the symmetry of an EVEN function. Name: 2. Describe the symmetry of an ODD function. 3. Describe each graph as EVEN, ODD, or NEITHER M. Winking Unit 3-5 page 52
4. Describe the definition in function notation of every EVEN function. 5. Describe a definition in function notation of every ODD function. 6. Describe each function below as EVEN, ODD, or NEITHER a. f(x) = x 2 + 5 b. g(x) = x 3 2x c. h(x) = x 5 4 d. m(x) = x 4 + 3x 2 + 2 e. p(x) = x f. q(x) = 3 7. If f(2) = 3 and f(x) is an EVEN function what other point must be on the graph of f(x)? 8. If g(2) = 3 and g(x) is an ODD function what other point must be on the graph of g(x)? 9. If the partially graphed function below is EVEN then finish what the rest of the graph should look like. 10. If the partially graphed function below is ODD then finish what the rest of the graph should look like. M. Winking Unit 3-5 page 53
Sec 3.6 Polynomial Functions Characteristics of Polynomial Functions Name: 1. Describe the Domain, Range, Intervals of Increase/Decrease, End Behavior, Intercepts. A. Consider the following function (Approximate) B. Consider the following function. (To the nearest tenth) i) Local Minimums: i) Local Minimums: ii)local Maximums: iii)describe the Domain: iv) Describe the Range: ii) Local Maximums: iii) Describe the Domain: iv) Describe the Range: v) Describe Intervals of Increase: v) Describe Intervals of Increase: vi) Describe Intervals of Decrease: vii) As x, determine f(x) viii) As x, determine f(x) ix) Determine the x-intercept: vi) Describe Intervals of Decrease: vii) As x decreases, determine f(x) viii) As x increases, determine f(x) ix) Determine the x-intercept: x) Determine the y-intercept: x) Determine the y-intercept: C. Sketch a graph of h(x). (Find all answers to the nearest tenth.) i) Local Minimums: ii)local Maximums: iii)describe the Domain: iv) Describe the Range: v) Describe Intervals of Increase: vi) Describe Intervals of Decrease: vii) As the graph moves left, f(x) viii) As the graph moves right, f(x) ix) Determine the x-intercept: x) Determine the y-intercept: M. Winking Unit 3-6 page 54
2. Determine the x-intercept and y-intercept of each of the following (use your graphing calculator for help). Round answers to the nearest hundredth when necessary. a. f(x) = x 4 + x 3 x 2 + x 2 b. g(x) = 2x 3 6x 2 + x + 3 y-intercept: x-intercept(s): y-intercept: x-intercept(s): 3. Based on the following partial set of table values of a polynomial function, determine between which two x-values you believe a zero may have occurred. a. b. 4. Based on the following partial set of table values of a polynomial function, determine between which two values you believe a local maximum or local minimum may have occurred. 5. a. b. 6. The following are graphs are of polynomial functions. Determine which of the following have an EVEN or ODD degree and whether the leading coefficient is POSITVE or NEGATIVE. a. b. c. d. The Degree of the Polynomial: EVEN or ODD Circle One The Leading Coefficient is: POSITIVE or NEGATIVE Circle One The Degree of the Polynomial: EVEN or ODD Circle One The Leading Coefficient is: POSITIVE or NEGATIVE Circle One M. Winking Unit 3-6 page 55 The Degree of the Polynomial: EVEN or ODD Circle One The Leading Coefficient is: POSITIVE or NEGATIVE Circle One The Degree of the Polynomial: EVEN or ODD Circle One The Leading Coefficient is: POSITIVE or NEGATIVE Circle One
7. Polynomial Functions in context. a. A baseball is struck such that its position on a plane perpendicular to home plate can be described by the equation y = 1 250 (x 175)2 + 121, where x is the horizontal distance from the plate in feet and y is the vertical distance also in feet. Determine the position of the ball when it is at its highest point. b. A company s net worth for the first 6 years after it opened can be modeled by the polynomial function: p(t) = t 3 6t 2 + 8t, where t is measure in years after the business opened and p(t) represent the company s net worth in millions. During what years did the company have a negative net worth? c. Many businesses know that most consumers associate price with quality (i.e. if it costs more it must be of a higher quality). An actuary at a cosmetics manufacture determined that the profit made the price charged for lipstick at a particular retail outlet could be modeled by: V(p) = 0.5p 3 + 12p 2 17p, where p is the price in dollars and V(p) is the amount earned by the retail store selling just the lipstick in a month. How much should the store charge to maximize its profit? d. A new hybrid car s fuel economy (MPG) depends on how fast the car is moving in MPH. The following polynomial function model is reasonable accurate from 20 mph to 90 mph. F(s) = 0.00002s 4 + 0.00356s 3 0.23022s 2 + 6.8832s 59.8, where s is the speed in miles per hour (MPH) and F(s) is the fuel economy in MPG. What speeds would between 20 and 90 mph would maximize fuel economy? M. Winking Unit 3-6 page 56
ANALYZING GRAPHS PROBLEM BUTTONS CALCULATOR Locate the maximum of the Function: y x 4 3x 3 2x 1. First graph the function Y =, (-), X,T,,n, ^, 4,, 3, X,T,,n, ^, 3, +, 2, X,T,,n, ZOOM, 6 2. Press, 2 nd, TRACE, 4 3. Using the LEFT arrow key,, move the cursor to the left of the maximum point and press ENTER. 4. Using the Right arrow key,, move the cursor to the right of the maximum point and press ENTER. 5. Finally, using the arrow keys,,, move the cursor to the approximate location of the maximum point and press ENTER. ( 2.14, 4.15) Locate the minimum of the Function: y x 2 5x Using a very similar technique try locating the minimum. Just use the MINIMUM command under the CALCULATE window. Matt Winking mmwinking@gmail.com