Explanation Given: The given polynomial function is f ( x ) = − x 4 + 2 x 3 . Concept: a) The leading term is the first term of a polynomial arranged in descending order of their powers. b) The leading coefficient is the coefficient of leading term. c) The degree is the highest power of a polynomial. d) The leading term behavior of curve for a polynomial of degree n can be explained in the following ways, 1. If n is an even number and a n > 0 , then end behavior of polynomial is given by the curve in Figure (1). Figure (1) 2. If n is an even number and a n < 0 , then end behavior of polynomial is given by the curve in Figure (2). Figure (2) 3. If n is an odd number and a n > 0 , then end behavior of polynomial is given by the curve in Figure (3). Figure (3) 4. If n is an odd number and a n < 0 , then end behavior of polynomial is given by the curve in Figure (4). Figure (4) e) According to the remainder theorem, the value of f ( x ) at x = c is equal to the remainder obtained by dividing f ( x ) by x − c , i.e. if, f ( x ) = ( x − c ) ⋅ Q ( x ) + R , then f ( c ) = R . Where, Q ( x ) is the quotient and R is the remainder obtained by dividing f ( x ) by x − c . f) The x -intercepts of a function is the point at which the graph of the function intersects the x -axis, or the value of x for f ( x ) = 0 . g) The y -intercepts of a function is the point at which the graph of the function intersects the y -axis, or the value of f ( x ) for x = 0 . Calculation: The given polynomial function is, f ( x ) = − x 4 + 2 x 3 f ( x ) = x 3 ( 2 − x ) ⋯ ( 1 ) Arrange the given polynomial function f ( x ) is in descending order of power, f ( x ) = − x 4 + 2 x 3 The first term of the given polynomial function is − x 4 . So, the leading term is − x 4 . The coefficient of leading term is − 1 . So the sign of leading coefficient is negative. The leading term has the highest power of 4. So, the degree of the polynomial is 4 that is even. Since, function is even with negative leading coefficient then the end behavior of the graph as x → ∞ and as x → − ∞ can be sketched as Substitute 0 for f ( x ) in equation (1). 0 = x 3 ( 2 − x ) x = 0 , 2 The zeros of f ( x ) are 0 and 2

5th Edition

ISBN: 9780321969576

Author: Judith A. Beecher, Judith A. Penna, Marvin L. Bittinger

Publisher: PEARSON

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Chapter 4.RE, Problem 16RE

To determine

To sketch:

The graph of the given polynomial function, f(x)=−x4+2x3.

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Chapter 4.RE, Problem 15RE

Chapter 4.RE, Problem 17RE

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Chapter 4 Solutions

College Algebra (5th Edition)

Ch. 4.1 - Determine the leading term, the leading...Ch. 4.1 - Prob. 2E Ch. 4.1 - Prob. 3E Ch. 4.1 - Prob. 4E Ch. 4.1 - Prob. 5E Ch. 4.1 - Prob. 6E Ch. 4.1 - Prob. 7E Ch. 4.1 - Prob. 8E Ch. 4.1 - Prob. 9E Ch. 4.1 - Prob. 10E

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The graph of the given polynomial function, f ( x ) = − x 4 + 2 x 3 .

To sketch: The graph of the given polynomial function, f(x)=−x4+2x3.

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Chapter 4 Solutions

To sketch:

The graph of the given polynomial function, f(x)=−x4+2x3.