Explanation Procedure used: The graph of a polynomial can sketch as follows. 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. We can also check with a graphing calculator. Calculation: The given polynomial function is f ( x ) = x ( x − 1 ) ( x + 3 ) ( x + 5 ) . 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 f ( x ) → ∞ as x → ∞ and x → − ∞ . Obtain the zeros of the function by solving f ( x ) = 0 . Any real zeros are the first coordinates of the x -intercepts. f ( x ) = 0 x ( x − 1 ) ( x + 3 ) ( x + 5 ) = 0 x = 0 or x − 1 = 0 or x + 3 = 0 or x + 5 = 0 x = 0 or x = 1 or x = − 3 or x = − 5 That is, the zeros of the functions are 0, 1, − 3 and − 5 . Thus, the x -intercepts are ( 0 , 0 ) , ( 1 , 0 ) , ( − 3 , 0 ) and ( − 5 , 0 ) . 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. Use the x -intercepts to divide the x axis into 5 intervals, ( − ∞ , − 5 ) , ( − 5 , − 3 ) , ( − 3 , 0 ) , ( 0 , 1 ) and ( 1 , ∞ ) . 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 ( − ∞ , − 5 ) ( − 5 , − 3 ) ( − 3 , 0 ) ( 0 , 1 ) ( 1 , ∞ ) Test value, x − 6 − 4 − 1 0

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Chapter 4.2, Problem 20E

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To graph: The polynomial function f(x)=x(x−1)(x+3)(x+5).

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Chapter 4.2, Problem 19E

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To graph: The polynomial function f ( x ) = x ( x − 1 ) ( x + 3 ) ( x + 5 ) .

Solution Summary: The author explains how the graph of a polynomial can sketch as follows: Obtain the zeros of the function by solving f(x)=0.

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To graph: The polynomial function f(x)=x(x−1)(x+3)(x+5).

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