MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial Division; The Remainder and Factor Theorems 4.4 Theorems about Zeros of Polynomial Functions 4.5 Rational Functions 4.6 Polynomial and Rational Inequalities Click the globe to the left and visit SAS Curriculum Pathways for interactive programs on Polynomial Functions. User: able7oxygen Quick Launch: 1022 (Polynomial patterns), 1441 (Exploring Graphs of Polynomial Functions) Pearson Interactive Figures Polynomial Leading Term Test 4.1 Polynomial Functions and Models Determine the behavior of the graph of a polynomial function using the leading term test. Factor polynomial functions and find the zeros and their multiplicities. Use a graphing calculator to graph a polynomial function and find its real number zeros. See the following lesson in Course Documents of CourseCompass: 171Session4 171Session4 ( Package file ) This lesson is a brief discussion of and suggestions relative to studying Chapter 4. You may use the "Polynomial Roots" program to graph polynomial functions and find the real roots (zeros). http://cfcc.edu/mathlab/geogebra/poly_roots.html Explanations and Exploratory Exercises Polynomial Function Quadratic Function A polynomial function P is given by where the coefficients a n, a n 1,, a 1, a 0 are real numbers and the exponents are whole numbers. 1
Cubic Function Examples of Polynomial Functions Examples of Nonpolynomial Functions Polynomial Functions The graph of a polynomial function is continuous and smooth. The domain of a polynomial function is the set of all real numbers 2
The Leading Term Test Example Using the leading term test, match each of the following functions with one of the graphs A D, which follow. a) b) c) d) Solution Finding Zeros of Factored Polynomial Functions If c is a real zero of a function (that is, f (c) = 0), then (c, 0) is an x intercept of the graph of the function. Graphs Example: Find the zeros of 3
Finding Zeros of Factored Polynomial Functions continued Solution: To solve the equation f(x) = 0, we use the principle of zero products, solving x 1 = 0 and x + 2 = 0. The zeros of f(x) are 1 and 2. Finding Real Zeros on a Calculator Find the zeros of f (x) = 0.2x 3 1.5x 2 0.3x + 2. Approximate the zeros to three decimal places. Solution Use a graphing calculator to create a graph. Look for points where the graph crosses the x axis. We use the ZERO feature to find them. See graph on right. The zeros are approximately 1.164, 1,142, and 7.523. http://cfcc.edu/mathlab/geogebra/poly_roots.html Even and Odd Multiplicity If (x c) k, k 1, is a factor of a polynomial function P(x) and (x c) k + 1 is not a factor and: k is odd, then the graph crosses the x axis at (c, 0); k is even, then the graph is tangent to the x axis at (c, 0). Example Find the zeros of f (x) = x 4 + 8x 2 33. Solution We factor as follows: f (x) = x 4 + 8x 2 33 = (x 2 + 11)(x 2 3). Solve the equation f(x) = 0 to determine the zeros. We use the principle of zero products. 4
307/2. Determine the leading term, the leading coefficient, and the degree of the polynomial. Then classify the polynomial function as constant, linear, quadratic, cubic, or quartic. f(x) = 15x 2 10 + 0.11x 4 7x 3 307/12. Select one of the following four sketches to describe the end behavior of the graph of the function. f(x) = (1/4)x 4 + (1/2)x 3 6x 2 + x 5 308/18. Select one of the following four sketches to describe the end behavior of the graph of the function. f(x) = 2x + x 3 5x 5 308/20. Use the leading term test to match the function with one of the graphs (a) (d), which follow. f(x) = 2x 4 x 2 + 1 5
308/24. Use substitution to determine whether 2, 3, and 1 are zeros of f(x) = 2x 3 3x 2 + x + 6 308/42. Find the zeros of the polynomial function and state the multiplicity of each f(x) = 3x 3 + x 2 48x 16 308/28. Find the zeros of the polynomial function and state the multiplicity of each f(x) = (x + 5) 3 (x 4)(x + 1) 2 6
308/46. Using a graphing calculator, find the real zeros of the function f(x) = x 4 2x 3 5.6 309/56. Using a graphing calculator, estimate the real zeros, the relative maxima and minima, and the range of the function f(x) = 2x 4 5.6x 2 + 10 309/58. Determine whether true or false: If P(x) = (x + 2) 2 (x 1/4) 5, then the graph of the polynomial function y = P(x) crosses the x axis at (1/4, 0). 309/62. Projectile Motion. A stone thrown downward with an initial velocity of 34.3 m/sec will travel a distance of s meters, where s(t) = 4.9t 2 + 34.3t and t is in seconds. If a stone is thrown downward at 34.3 m/sec from a height of 294 m, how long will it take the stone to hit the ground? 7
310/66. Windmill Power. Under certain conditions, the power P, in watts per hour, generated by a windmill with winds blowing v miles per hour is given by P(v) = 0.015v 3. (a) Find the power generated by 15 mph winds. (b) How fast must the wind blow in order to generate 120 watts of power in 1 hr? 310/70. Threshold Weight. In a study performed by Alvin Shemesh, it was found that the threshold weight W, defined as the weight above which the risk of death rises dramatically, is given by W(h) = (h/12.3) 3, where W is in pounds and h is a persons height, in inches. Find the threshold weight of a person who is 5 ft 7 in. tall. 310/74. Determine which, if any, of the following functions might be used as a model for the data. a) Linear, f(x) = mx + b b) Quadratic, f(x) = ax 2 + bx + c, a > 0 c) Quadratic, f(x) = ax 2 + bx + c, a < 0 d) Polynomial, not quadratic or linear 311/79. Unemployed. The table below shows the number of unemployed in the United States from 1996 through 2006. a) Use a graphing calculator to fit cubic and quartic functions to the data. Let x represent the number of years since 1996. b) Use the functions found in part (a) to estimate the number of unemployed in 2008. Compare the estimates and determine which model gives the more realistic estimate. R 2 = 0.8366 See TI tutorial at http://cfcc.edu/faculty/cmoore/ti83modeling.htm R 2 = 0.9145 See TI tutorial at http://cfcc.edu/faculty/cmoore/ti83modeling.htm See TI tutorial at http://cfcc.edu/faculty/cmoore/ti83modeling.htm 8