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 3 + 3 x 2 + 6 x − 8 . Use the leading term test to determine the end behavior. The leading term in the given polynomial is x 3 . So, the degree of the polynomial is 3, which is an odd number. The coefficient of the leading term is 1, which is a positive number. So, it is clear that the value of f ( x ) → ∞ as x → − ∞ and f ( x ) → − ∞ as 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 3 + 3 x 2 + 6 x − 8 = 0 Find the zeros of the above polynomial by using synthetic division. Check x = 1 is the zero of the polynomial function f ( x ) = − x 3 + 3 x 2 + 6 x − 8 . 1 − 1 3 6 − 8 − 1 2 8 _ − 1 2 8 0 From the above synthetic division, f ( 1 ) = 0 . Therefore, x − 1 is a factor of the polynomial function f ( x ) = − x 3 + 3 x 2 + 6 x − 8 . Find the remaining zeros of the polynomial as follows, it is enough to solve f ( x ) = 0 . − x 3 + 3 x 2 + 6 x − 8 = 0 ( x − 1 ) ( − x 2 + 2 x + 8 ) = 0 ( x − 1 ) ( − x 2 + 4 x − 2 x + 8 ) = 0 On further simplifications, ( x − 1 ) ( − x ( x − 4 ) − 2 ( x − 4 ) ) = 0 ( x − 1 ) ( x − 4 ) ( − ( x + 2 ) ) = 0 x − 1 = 0 or x − 4 = 0 or − ( x + 2 ) = 0 x = 1 or x = 4 or x = − 2 Thus, the zeros of the functions are , –2, 1 and 4

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Chapter 4.3, Problem 53E

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To graph: The polynomial function f(x)=−x3+3x2+6x−8.

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Chapter 4.3, Problem 52E

Chapter 4.3, Problem 54E

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To graph: The polynomial function f ( x ) = − x 3 + 3 x 2 + 6 x − 8 .

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To graph: The polynomial function f(x)=−x3+3x2+6x−8.

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Chapter 4 Solutions