1 CA-Fall 2011-Jordan College Algebra, 4 th edition, Beecher/Penna/Bittinger, Pearson/Addison Wesley, 2012 Chapter 4: Polynomial Functions and Rational Functions Section 4.1 Polynomial Functions and Models Polynomial Function Polynomial Function Degree Example Constant 0 f(x) = 4 Linear 1 f(x) = 3x + 1 Quadratic 2 f(x) = 4x 2 x + 9 Cubic 3 f(x) = x 3 + 2x 2 x + 11 Quartic 4 f(x) = x 4 3.2x 3 + 0.1x Example 1 f(x) = 15x 2 10 + 0.11x 4 7x 3 a) determine the leading term b) determine the leading coefficient c) determine the degree of the polynomial d) classify the polynomial as constant, linear, quadratic, cubic, or quartic Characteristics of Polynomial Functions Graphs of polynomial functions are with no holes or breaks. Graphs of polynomial functions are with no sharp corners. The domain of a polynomial function is the set of all real numbers, (-, ).
2 The Leading-Term Test If a n x n is the leading term of a polynomial function, then the end behavior of the graph as x or as x - can be described in one of the following four ways. If n is even and a n > 0: If n is even and a n < 0: (both ends will rise) (both ends will fall) If n is odd and a n > 0: If n is odd and a n < 0: (fall to left and rise to right) (rise to left and fall to right) Example 2 Sketch an end behavior diagram to describe the end behavior of the graph of the function f(x) = -6x 3 4x 2 + 2x 1
Example 3 Sketch an end behavior diagram to describe the end behavior of the graph of the function f(x) = -5 + x 6x 2 + ½ x 3 + ¼ x 4 3 Factor/Zero/x-intercept If (x a) is a factor, then x = a is a zero and (a, 0) is an x-intercept. Even and Odd Multiplicity The number of times an x-value is a zero in a function is called its multiplicity. If an x-value is a zero an odd number of times in a function, then the zero is said to have odd multiplicity and the graph will the x-axis at that x-intercept. If an x-value is a zero an even number of times in a function, then the zero is said to have even multiplicity and the graph will be tangent to (, but not cross) the x-axis at that x-intercept. Example 4 Example 5 Find the zeros of the polynomial function and state the multiplicity of each. f(x) = (x + 5) 3 (x 4) (x + 1) 2 Find the zeros of the polynomial function and state the multiplicity of each. f(x) = x 4 + 8x 2 33 Using a Graphing Calculator to Approximate a Zero Type the function in Y= and press ZOOM 6. 2 nd TRACE 2: zero (or 2: root) Move the cursor to the left of the zero and then press ENTER. You may be either above or below the x-axis when you are to the left of the zero. Don t focus on that use the left/right arrow keys to move. Move the cursor to the right of the zero and then press ENTER. Again you may be either above or below the x-axis when you are to the right of the zero. Only use the left/right arrow keys. Press ENTER when the calculator says Guess? The bottom of the calculator screen will say Zero (or Root). The answer will be the x-value. The y-value should be zero. Do this process for as many zeros as there are on the graph. Example 6 Using a graphing calculator, find the real zeros of the function to three decimal places. f(x) = x 4 2x 3 5.6
Section 4.2 Graphing Polynomial Functions 4 Maximum Number of Real Zeros and x-intercepts If P(x) is a polynomial function of degree n, then the graph of the function has at most n real zeros. This also means that the graph will have at most n x-intercepts. Maximum Number of Turnings Points A turning point of a graph is a relative maximum or a relative minimum (peaks and valleys). If P(x) is a polynomial function of degree n, then the graph will have at most turning points. Example 1 If f(x) = -x 2 + x 4 x 6 + 3, find a) the maximum number of real zeros that the function can have b) the maximum number of x-intercepts that the graph of the function can have c) the maximum number of turning points that the graph of the function can have Graphing Polynomial Functions Use the leading-term test to determine the end behavior. Find the real zeros by solving f(x) = 0. Any real zeros are the first coordinates of the x-intercepts of the graph. Determine the multiplicities of the zeros to determine if the graph crosses the x-axis at the x-intercept or if the graph is tangent to the x-axis at the x-intercept. Find the y-intercept by finding f(0). Start the graph by plotting all the intercepts. Then, pick an x-value between and around all x-intercepts and put these x-values in a table. You may use the TABLE function on your calculator to fill in the corresponding y-values in the table. Plot these points from your table and draw the general shape of the graph. Make sure your graph passes the vertical line test. As a partial check, use the facts that the graph has at most n x-intercepts and at most n 1 turning points. Example 2 Graph the polynomial function f(x) = x 4 4x 3 + 3x 2 Example 3 Graph the polynomial function f(x) = -x(x 3)(x 3)(x + 2)
Section 4.3 Polynomial Division; The Remainder Theorem and the Factor Theorem 5 Division Algorithm dividend = (divisor)(quotient) + remainder P(x) = d(x) Q(x) + R(x) Review Long Division Example 1 Use long division to find the quotient Q(x) and the remainder R(x) when P(x) is divided by d(x) and express P(x) in the form P(x) = d(x) Q(x) + R(x). P(x) = x 3 9x 2 + 15x + 25 and d(x) = x 5 Synthetic Division Synthetic division is a collapsed version of long division where only the coefficients of the terms are written. Synthetic division can only be used when the divisor is of the form x k (linear). If any terms are missing in the dividend, then a must hold their place. Example 2 Use synthetic division to find the quotient and the remainder. (-4x 5 + x 4 + 6x 3 + 2x 2 + 50) (x 2) Remainder Theorem If a polynomial f(x) is divided by x k, the remainder is f(k). In other words, the remainder is equal to the y-value. Example 3 Use synthetic division to find the function values. f(x) = 2x 4 + x 2 10x + 1 a) Find f(-10) b) Find f(3) Example 4 Use synthetic division to determine whether the given number is a zero of the polynomial function. -1; g(x) = x 4 6x 3 + x 2 + 24x 20 Example 5 Use synthetic division to determine whether the given number is a zero of the polynomial function. -2i; g(x) = x 3 4x 2 + 4x 16
Factor Theorem For a polynomial f(x), if f(c) = 0, then x c is a factor of f(x). In other words, if the y-value is 0 then the remainder is which means that the divisor is a factor. Example 6 Factor the polynomial f(x). Then solve the equation f(x) = 0. f(x) = x 3 3x 2 10x + 24 6
Section 4.4 Theorems about Zeros of Polynomial Functions 7 The Fundamental Theorem of Algebra Every polynomial function of degree n, with n 1, has at least one zero in the system of complex numbers. As a result, every nth degree polynomial has n zeros and n linear factors although they may not be unique. Example 1 Find a polynomial function of degree 4 having zeros 1, 2, 4i, and -4i. Nonreal Zeros If a polynomial has real coefficients, then complex zeros occur in conjugate pairs. For example, if -2 + 3i is a zero, then is also a zero. Irrational Zeros If a polynomial has rational coefficients, then irrational zeros occur in conjugate pairs. For example, if 2 3 is a zero, then is also a zero. Example 2 Suppose that a polynomial function of degree 4 with rational coefficients has the given numbers as zeros. Find the other zero(s)., -1, 4/5 Example 3 Suppose that a polynomial function of degree 6 with rational coefficients has -3 + 2i, -6i, and as three of its zeros. Find the other zero(s). Example 4 Find a polynomial function of lowest degree with rational coefficients that has the given numbers as some of its zeros: 3, 5i Example 5 Suppose the polynomial function f(x) = x 4 5x 3 + 7x 2 5x + 6 has i as one of its zeros. Find the other zeros.
Rational Zeros Theorem To obtain a list of possible rational zeros of a polynomial, divide the factors of the constant term by the factors of the leading coefficient. 8 Example 6 List all possible rational zeros of the function f(x) = 10x 25 + 3x 17 35x + 5. Example 7 Given f(x) = 4x 3 18x 2 + 24x + 17, a) find the rational zeros and then the other zeros; that is, solve f(x) = 0 b) factor f(x) into linear factors Example 8 Given f(x) = 5x 4 4x 3 + 44x 2 36x 9, a) find the rational zeros and then the other zeros; that is, solve f(x) = 0 b) factor f(x) into linear factors
Section 4.5 Rational Functions 9 Vertical Asymptotes The vertical asymptote is a vertical line that the graph never touches nor crosses. The vertical asymptote of a rational function is found by setting the equal to zero and solving for the variable. This assumes that there are no common factors other than constants between the numerator and denominator of the rational function. Example 1 Determine the vertical asymptotes for the graphs of the functions. a) b) Horizontal Asymptotes A horizontal asymptote is a horizontal line that the graph of the function approaches as the graph continues to the left or to the right or both. The graph of a rational function might cross a horizontal asymptote. To find the horizontal asymptote, look at the degrees of the numerator and denominator. 1. degree of numerator < degree of denominator y = 0 2. degree of numerator = degree of denominator y = leading coefficient of numerator divided by leading coefficient of denominator 3. degree of numerator > degree of denominator no horizontal asymptote Example 2 Determine the horizontal asymptotes for the graphs of the functions. a) b) c) Oblique Asymptotes An oblique asymptote is a slanted line that the graph of the function approaches as the graph continues to the left or to the right or both. An oblique asymptote occurs when the degree of the numerator is greater than the degree of the denominator.
Graphing Rational Functions Find any vertical asymptotes. 10 Find any horizontal asymptotes. Find the zeros of the function (x-intercepts). Find the y-intercept of the function. Dash the asymptotes and plot the intercepts. Create a table of ordered pairs by picking x-values between and around all the intercepts and the vertical asymptote. Plot the points from the table and sketch the graph, using the asymptotes as guides. Example 3 Graph the rational function Example 4 Graph the rational function Applications of Horizontal Asymptotes Example 5 The average cost per disc, in dollars, for a company to produce x DVDs on exercising is given by the function, x > 0. a) Find the horizontal asymptote of the graph and complete the following: A(x) as x. b) Explain the meaning of the answer to part (a) in terms of the application.
Section 4.6 Polynomial Inequalities 11 Steps for Solving a Polynomial Inequality Set one side of the inequality to zero. Factor (or use the quadratic formula if applicable) to find the zeros of the polynomial. Mark the real zeros on a number line. Select a test value from each interval and plug that value into the inequality to obtain either a True or a False statement. The solution to the inequality will be the interval(s) that make the inequality true. Write the solution using interval notation. Example 1 Solve (x 2)(x + 1) 0 Example 2 Solve x 2 x 20 < 0 Example 3 Solve 4x x 2 > 12 Example 4 Solve x 3 + x 6 4x 2