Explanation Procedure used: The graph of a polynomial, follow the below steps. 1. Use the leading term test to determine the end behavior. 2. Find the zeros of the function by solving f ( x ) = 0 . Any real zeros are the first coordinates of the x -intercepts. 3. Use the x -intercepts (zeros) to divide the x -axis into the intervals, and choose a test point in each interval to determine the sign of all function values in that interval. 4. Find f ( 0 ) . This gives the y -intercept of the function. 5. If necessary, find additional function values to determine the general shape of the graph and then draw the graph. 6. As a partial check, use the facts that the graph has at most n x -intercepts and at most n − 1 turning points. Multiplicity of zeros can also be considered in order to check where the graph crosses or is tangent to the x -axis. Calculation: The given polynomial function is h ( x ) = x ( x − 4 ) ( x + 1 ) ( x − 2 ) . The leading term in the given polynomial is x 4 . So, the degree of the polynomial is 4, which is an even number. The coefficient of the leading term is 1 , which is a positive number. So, it is observed that the value of h ( x ) → ∞ as x → ∞ and h ( x ) → ∞ as x → − ∞ . Obtain the zeros of the function by solving h ( x ) = 0 . Any real zeros are the first coordinates of the x -intercepts. h ( x ) = 0 x ( x − 4 ) ( x + 1 ) ( x − 2 ) = 0 x = 0 or x − 4 = 0 or x + 1 = 0 or x − 2 = 0 x = 0 or x = 4 or x = − 1 or x = 2 That is, the zeros of the functions are 0, 4, − 1 and 2. Thus, the x -intercepts are ( 0 , 0 ) , ( 4 , 0 ) , ( − 1 , 0 ) and ( 2 , 0 ) . Use the x -intercepts to divide the x axis into 5 intervals, ( − ∞ , − 1 ) , ( − 1 , 0 ) , ( 0 , 2 ) , ( 2 , 4 ) and ( 4 , ∞ ) . Now, choose some points for x from each of these intervals to obtain the sign of the function. Tabulate the values and the signs as shown below in Table 1. Interval ( − ∞ , − 1 ) ( − 1 , 0 ) ( 0 , 2 ) ( 2 , 4 ) ( 4 , ∞ ) Test value, x − 2 − 0

College Algebra: Graphs and Models (6th Edition)

6th Edition

ISBN: 9780134179032

Author: Marvin L. Bittinger, Judith A. Beecher, David J. Ellenbogen, Judith A. Penna

Publisher: PEARSON

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Chapter 4.2, Problem 19E

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To graph: The polynomial function h(x)=x(x−4)(x+1)(x−2).

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Chapter 4.2, Problem 18E

Chapter 4.2, Problem 20E

Chapter 4 Solutions

College Algebra: Graphs and Models (6th Edition)

Ch. 4.1 - Determine the leading term, the leading...Ch. 4.1 - Prob. 2E Ch. 4.1 - Determine the leading term, the leading...Ch. 4.1 - Prob. 4E Ch. 4.1 - Prob. 5E Ch. 4.1 - Determine the leading term, the leading...Ch. 4.1 - Prob. 7E Ch. 4.1 - Prob. 8E Ch. 4.1 - Determine the leading term, the leading...Ch. 4.1 - Prob. 10E

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To graph: The polynomial function h ( x ) = x ( x − 4 ) ( x + 1 ) ( x − 2 ) .

Solution Summary: The author explains how to graph a polynomial by using the leading term test to determine the end behavior and finding the zeros of the function.

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To graph: The polynomial function h(x)=x(x−4)(x+1)(x−2).

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