1. Copyright © 2000 by the McGraw-Hill Companies, Inc. Barnett/Ziegler/Byleen Precalculus: A Graphing Approach Chapter Three Polynomial & Rational Functions

2. Copyright © 2000 by the McGraw-Hill Companies, Inc. Synthetic Division 3-1-23 2x 3 – 1x 2 + 2x – 5 Quotient Dividend coefficients Quotient coefficientsRemainder

3. Copyright © 2000 by the McGraw-Hill Companies, Inc. P(x) = a n x n + a n–1 x n–1 +... + a 1 x + a 0, a n  0 1. a n > 0 and n even Graph of P(x) increases without bound as x decreases to the left and as x increases to the right. P(x)   as x  –  P(x)   as x   2. a n > 0 and n odd Graph of P(x) decreases without bound as x decreases to the left and increases without bound as x increases to the right. P(x)  –  as x  –  P(x)   as x   Left and Right Behavior of a Polynomial 3-1-24(a) (x) x y ) x y

4. Copyright © 2000 by the McGraw-Hill Companies, Inc. Left and Right Behavior of a Polynomial P(x) = a n x n + a n–1 x n–1 +... + a 1 x + a 0, a n  0 3. a n < 0 and n even Graph of P(x) decreases without bound as x decreases to the left and as x increases to the right. P(x)  –  as x  –  P(x)  –  as x   4. a n < 0 and n odd Graph of P(x) increases without bound as x decreases to the left and decreases without bound as x increases to the right. P(x)   as x  –  P(x)  –  as x   3-1-24(b) x y ) x y x)x)

5. Copyright © 2000 by the McGraw-Hill Companies, Inc. Fundamental Theorem of Algebra Every polynomial P(x) of degree n > 0 has at least one zero. n Zeros Theorem Every polynomial P(x) of degree n > 0 can be expressed as the product of n linear factors. Hence, P(x) has exactly n zeros—not necessarily distinct. Imaginary Zeros Theorem Imaginary zeros of polynomials with real coefficients, if they exist, occur in conjugate pairs. Real Zeros and Odd-Degree Polynomials A polynomial of odd degree with real coefficients always has at least one real zero. 3-2-25

6. Copyright © 2000 by the McGraw-Hill Companies, Inc. Rational Zero Theorem If the rational number in lowest terms, is a zero of the polynomial P(x) = a n x n + a n-1 x n-1 +... + a 1 x + a 0 a n  0 with integer coefficients, then b must be an integer factor of a 0 and c must be an integer factor of a n. 3-2-26

7. Copyright © 2000 by the McGraw-Hill Companies, Inc. If f is continuous on an interval I, a and b are two numbers in I, and f(a) and f(b) are of opposite sign, then there is at least one x intercept between a and b. Given an nth-degree polynomial P(x) with real coefficients, n > 0, a n > 0, and P(x) divided by x – r using synthetic division: 1. Upper Bound. If r > 0 and all numbers in the quotient row of the synthetic division, including the remainder, are nonnegative, then r is an upper bound of the real zeros of P(x). 2. Lower Bound. If r < 0 and all numbers in the quotient row of the synthetic division, including the remainder, alternate in sign, then r is a lower bound of the real zeros of P(x). [Note: In the lower-bound test, if 0 appears in one or more places in the quotient row, including the remainder, the sign in front of it can be considered either positive or negative, but not both. For example, the numbers 1, 0, 1 can be considered to alternate in sign, while 1, 0, –1 cannot.] Location Theorem 3-3-27 Upper and Lower Bounds of Real Zeros

8. Copyright © 2000 by the McGraw-Hill Companies, Inc. Approximate to one decimal place the zero of P(x) = x 4 – 2x 3 – 10x 2 + 40x – 90 on the interval (3, 4). 343.53.75 3.6253.5625 (( ) ( ) ) x Nested intervals produced by the Bisection Method The Bisection Method 3-3-28

9. Copyright © 2000 by the McGraw-Hill Companies, Inc. Zeros of Even and Odd Multiplicity If P(x) is a polynomial with real coefficients, then: 1.If r is a zero of odd multiplicity, then P(x) changes sign at r and does not have a local extremum at x = r. 2.If r is a zero of even multiplicity, then P(x) does not change sign at r and has a local extremum at x = r. The bisection method requires that the function change sign at a zero in order to approximate that zero. Thus, this method will always fail at a zero of even multiplicity. Zeros of even multiplicity can be approximated by using a maximum or minimum approximation routine, whichever applies. 3-3-29

10. Copyright © 2000 by the McGraw-Hill Companies, Inc. Step 1.Intercepts. Find the real solutions of the equation n(x) = 0 and use these solutions to plot any x intercepts of the graph of f. Evaluate f(0), if it exists, and plot the y intercept. Step 2.Vertical Asymptotes. Find the real solutions of the equation d(x) = 0 and use these solutions to determine the domain of f, the points of discontinuity, and the vertical asymptotes. Sketch any vertical asymptotes as dashed lines. Step 3.Horizontal Asymptotes. Determine whether there is a horizontal asymptote, and if so, sketch it as a dashed line. Step 4.Complete the Sketch. Using a graphing utility graph as an aid, and the information determined in steps 1-3, sketch the graph. Analyzing and Sketching the Graph of a Rational Function: 3-4-30