To determine To graph: The polynomial function g ( x ) = x 4 − 4 x 3 + 3 x 2 . 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 g ( x ) = x 4 − 4 x 3 + 3 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 g ( x ) → ∞ as x → ∞ and x → − ∞ . Obtain the zeros of the function by solving g ( x ) = 0 . Any real zeros are the first coordinates of the x -intercepts. g ( x ) = 0 x 4 − 4 x 3 + 3 x 2 = 0 x 2 ( x 2 − 4 x + 3 ) = 0 x 2 ( x − 1 ) ( x − 3 ) = 0 On further simplification, x 2 ( x − 1 ) ( x − 3 ) = 0 x 2 = 0 or x − 1 = 0 or x − 3 = 0 x = 0 or x = 1 or x = 3 That is, the zeros of the functions are 0, 1 and 3 . Thus, the x -intercepts will be ( 0 , 0 ) , ( 1 , 0 ) and ( 3 , 0 ) . Use the x -intercepts to divide the x axis into 4 intervals, ( − ∞ , 0 ) , ( 0 , 1 ) , ( 1 , 3 ) and ( 3 , ∞ ) . Now, choose some points for x from each of these intervals to obtain the sign of the function. Substitute x = − 1 in g ( x ) = x 4 − 4 x 3 + 3 x 2 . g ( − 1 ) = ( − 1 ) 4 − 4 ( − 1 ) 3 + 3 ( − 1 ) 2 = 1 + 4 + 3 = 8 Substitute x = 0.5 in g ( x ) = x 4 − 4 x 3 + 3 x 2 . g ( 0.5 ) = ( 0.5 ) 4 − 4 ( 0.5 ) 3 + 3 ( 0.5 ) 2 = 0

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Chapter 4.2, Problem 14E

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To graph: The polynomial function g(x)=x4−4x3+3x2.

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Chapter 4.2, Problem 13E

Chapter 4.2, Problem 15E

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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 - Determine the leading term, the leading...Ch. 4.1 - Determine the leading term, the leading...Ch. 4.1 - Prob. 6E Ch. 4.1 - Determine the leading term, the leading...Ch. 4.1 - Prob. 8E Ch. 4.1 - Prob. 9E Ch. 4.1 - Prob. 10E

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