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Chapter 2: Problem 19

Use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. $$f(x)=5 x^{3}+7 x^{2}-x+9$$

Short Answer

Expert verified
The end behavior is: as x approaches negative infinity, f(x) approaches negative infinity; and as x approaches positive infinity, f(x) approaches positive infinity.

Step by step solution

01

Identify the Degree of the Polynomial

The degree of a polynomial is the highest power of x in the function. In the function \( f(x)=5 x^{3}+7 x^{2}-x+9\), the highest power of x is 3, so the degree is 3.
02

Identify the Leading Coefficient

The leading coefficient is the coefficient of the term with the highest power of x. In the function \(f(x)=5 x^{3}+7 x^{2}-x+9\), the coefficient of the x^3 term is 5, so the leading coefficient is 5.
03

Apply the Leading Coefficient Test

The Leading Coefficient Test tells us that if the degree of the polynomial is odd and the leading coefficient is positive, then the end behavior of the graph will be: as x approaches negative infinity, f(x) approaches negative infinity, and as x approaches positive infinity, f(x) approaches positive infinity. Therefore, given that the degree of our polynomial is 3 (odd) and the leading coefficient is 5 (positive), we conclude that as x tends to negative infinity, f(x) tends to negative infinity; as x tends to positive infinity, f(x) tends to positive infinity.

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