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- 1. MATH 107 Section 3.2 Polynomial Functions
- 2. 2© 2010 Pearson Education, Inc. All rights reserved Definitions A polynomial function of degree n is a function of the form where n is a nonnegative integer and the coefficients an, an–1, …, a2, a1, a0 are real numbers with an ≠ 0. f x( ) = an xn + an-1xn-1 + ...+ a2 x2 + a1x + a0 ,
- 3. 3© 2010 Pearson Education, Inc. All rights reserved Definitions A constant function f (x) = a, (a ≠ 0) which may be written as f (x) = ax0, is a polynomial of degree 0. The term anxn is called the leading term. The number an is called the leading coefficient, and a0 is the constant term.
- 4. 4© 2010 Pearson Education, Inc. All rights reserved Definitions Degree Name 0 Zero function: f(x)=0 1 linear 2 quadratic 3 cubic 4 quartic 5 quintic
- 5. 5© 2010 Pearson Education, Inc. All rights reserved COMMON PROPERTIES OF POLYNOMIAL FUNCTIONS 1. The domain of a polynomial function is the set of all real numbers.
- 6. 6© 2010 Pearson Education, Inc. All rights reserved 2. The graph of a polynomial function is a continuous curve.
- 7. 7© 2010 Pearson Education, Inc. All rights reserved 3. The graph of a polynomial function is a smooth curve.
- 8. 8© 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Polynomial Functions State which functions are polynomial functions. For each polynomial function, find its degree, the leading term, and the leading coefficient. f (x) = 5x4 – 2x + 7 Solution
- 9. 9© 2010 Pearson Education, Inc. All rights reserved END BEHAVIOR OF POLYNOMIAL FUNCTIONS Case 1 n Even a > 0 The graph rises to the left and right, similar to y = x2.
- 10. 10© 2010 Pearson Education, Inc. All rights reserved END BEHAVIOR OF POLYNOMIAL FUNCTIONS Case 2 n Even a < 0 The graph falls to the left and right, similar to y = –x2.
- 11. 11© 2010 Pearson Education, Inc. All rights reserved END BEHAVIOR OF POLYNOMIAL FUNCTIONS Case 3 n Odd a > 0 The graph rises to the right and falls to the left, similar to y = x3.
- 12. 12© 2010 Pearson Education, Inc. All rights reserved END BEHAVIOR OF POLYNOMIAL FUNCTIONS Case 4 n Odd a < 0 The graph rises to the left and falls to the right, similar to y = –x3.
- 13. 13© 2010 Pearson Education, Inc. All rights reserved EXAMPLE 2 Understanding the End Behavior of a Polynomial Function Let function of degree 3. Show that P x( ) = 2x3 + 5x2 - 7x +11 be a polynomial 3 2P x x when |x| is very large. Solution P x( ) = x3 2 + 5 x - 7 x2 + 11 x3 æ èç ö ø÷ When |x| is very large 5 x , 7 x2 and 11 x3 are close to 0. P x( ) » x3 2 + 0 - 0 + 0( ) » 2x3 .Therefore,
- 14. 14© 2010 Pearson Education, Inc. All rights reserved THE LEADING-TERM TEST Its leading term is anxn. The behavior of the graph of f as x → ∞ or as x → –∞ is similar to one of the following four graphs and is described as shown in each case. The middle portion of each graph, indicated by the dashed lines, is not determined by this test. Let 1 1 1 0... 0n n nn nf x a x ax ax aa be a polynomial function.
- 15. 15© 2010 Pearson Education, Inc. All rights reserved Case 1 n Even an > 0 THE LEADING-TERM TEST
- 16. 16© 2010 Pearson Education, Inc. All rights reserved Case 2 n Even an < 0 THE LEADING-TERM TEST
- 17. 17© 2010 Pearson Education, Inc. All rights reserved Case 3 n Odd an > 0 THE LEADING-TERM TEST
- 18. 18© 2010 Pearson Education, Inc. All rights reserved Case 4 n Odd an < 0 THE LEADING-TERM TEST
- 19. 19© 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Using the Leading-Term Test Use the leading-term test to determine the end behavior of the graph of y = f x( ) = -2x3 + 3x2 + 4. Solution Here n = 3 (odd) and an = –2 < 0. Thus, Case 4 applies. The graph of f (x) rises to the left and falls to the right. This behavior is described as y ∞ as x –∞ and y –∞ as x ∞.
- 20. 20© 2010 Pearson Education, Inc. All rights reserved REAL ZEROS OF POLYNOMIAL FUNCTIONS 1. c is a zero of f . 2. c is a solution (or root) of the equation f (x) = 0. 3. c is an x-intercept of the graph of f . The point (c, 0) is on the graph of f . If f is a polynomial function and c is a real number, then the following statements are equivalent.
- 21. 21© 2010 Pearson Education, Inc. All rights reserved EXAMPLE 4 Finding the Zeros of a Polynomial Function Find all zeros of each polynomial function. 3 2 3 2 a. 2 2 b. 2 2 f x x x x g x x x x Solution Factor f (x) and then solve f (x) = 0.
- 22. 22© 2010 Pearson Education, Inc. All rights reserved REAL ZEROS OF POLYNOMIAL FUNCTIONS A polynomial function of degree n with real coefficients has, at most, n real zeros.
- 23. 23© 2010 Pearson Education, Inc. All rights reserved EXAMPLE 6 Finding the Number of Real Zeros Find the number of distinct real zeros of the following polynomial functions of degree 3. Solution 22 a. 1 2 3 b. 1 1 c. 3 1 f x x x x g x x x h x x x
- 24. 24© 2010 Pearson Education, Inc. All rights reserved MULTIPLICITY OF A ZERO If c is a zero of a polynomial function f (x) and the corresponding factor (x – c) occurs exactly m times when f (x) is factored, then c is called a zero of multiplicity m. m Behavior of f at x=c Odd Crosses Even touches
- 25. 25© 2010 Pearson Education, Inc. All rights reserved ODD MULTIPLICITY OF A ZERO
- 26. 26© 2010 Pearson Education, Inc. All rights reserved EVEN MULTIPLICITY OF A ZERO
- 27. 27© 2010 Pearson Education, Inc. All rights reserved EXAMPLE 7 Finding the Zeros and Their Multiplicity Find the zeros of the polynomial function f (x) = x2(x + 1)(x – 2), and give the multiplicity of each zero. Solution
- 28. 28© 2010 Pearson Education, Inc. All rights reserved TURNING POINTS A local (or relative) maximum value of f is higher than any nearby point on the graph. A local (or relative) minimum value of f is lower than any nearby point on the graph. The graph points corresponding to the local maximum and local minimum values are called turning points. At each turning point the graph changes from increasing to decreasing or vice versa.
- 29. 29© 2010 Pearson Education, Inc. All rights reserved TURNING POINTS The graph of f has turning points at (–1, 12) and at (2, –15). f x( ) = 2x3 - 3x2 -12x + 5
- 30. 30© 2010 Pearson Education, Inc. All rights reserved NUMBER OF TURNING POINTS If f (x) is a polynomial of degree n, then the graph of f has, at most, (n – 1) turning points.
- 31. 31© 2010 Pearson Education, Inc. All rights reserved EXAMPLE 8 Finding the Number of Turning Points Use a graphing calculator and the window –10 x 10; –30 y 30 to find the number of turning points of the graph of each polynomial. 4 2 3 2 3 2 a. 7 18 b. 12 c. 3 3 1 f x x x g x x x x h x x x x
- 32. 32© 2010 Pearson Education, Inc. All rights reserved EXAMPLE 8 Finding the Number of Turning Points Solution f has three total turning points; two local minimum and one local maximum. a. f x( ) = x4 - 7x2 -18
- 33. 33© 2010 Pearson Education, Inc. All rights reserved EXAMPLE 8 Finding the Number of Turning Points Solution continued g has two total turning points; one local maximum and one local minimum. b. g x( ) = x3 + x2 -12x
- 34. 34© 2010 Pearson Education, Inc. All rights reserved EXAMPLE 8 Finding the Number of Turning Points Solution continued h has no turning points, it is increasing on the interval (–∞, ∞). c. h x( ) = x3 - 3x2 + 3x -1
- 35. 35© 2010 Pearson Education, Inc. All rights reserved GRAPHING A POLYNOMIAL FUNCTION Step 1 Determine the end behavior. Apply the leading-term test. Step 2 Find the zeros of the polynomial function. Set f (x) = 0 and solve. The zeros give the x-intercepts. Step 3 Find the y-intercept by computing f (0).
- 36. 36© 2010 Pearson Education, Inc. All rights reserved Step 4 Draw the graph. Use the multiplicities of each zero to decide whether the graph crosses the x-axis. Use the fact that the number of turning points is less than the degree of the polynomial to check whether the graph is drawn correctly.
- 37. 37© 2010 Pearson Education, Inc. All rights reserved EXAMPLE 9 Graphing a Polynomial Function Sketch the graph of 3 2 4 4 16.f x x x x Solution Step 1 Determine end behavior. Degree = 3 Leading coefficient = –1 End behavior shown in sketch.
- 38. 38© 2010 Pearson Education, Inc. All rights reserved EXAMPLE 9 Graphing a Polynomial Function Solution continued Step 2 Find the zeros by setting f (x) = 0. Each zero has multiplicity 1, the graph crosses the x-axis at each zero.
- 39. 39© 2010 Pearson Education, Inc. All rights reserved EXAMPLE 9 Graphing a Polynomial Function Solution continued Step 3 Find the y-intercept by computing f (0). The y-intercept is f (0) = 16. The graph passes through (0, 16).
- 40. 40© 2010 Pearson Education, Inc. All rights reserved EXAMPLE 9 Graphing a Polynomial Function Solution continued Step 4 Draw the graph. The number of turning points is 2, which is less than 3, the degree of f.

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