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Chapter 2: Problem 22

Use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. $$f(x)=11 x^{4}-6 x^{2}+x+3$$

Short Answer

Expert verified
The end behavior of the polynomial function \(f(x) = 11x^{4}-6x^{2}+x+3\) is that as \(x\) goes to both positive and negative infinity, \(f(x)\) heads to positive infinity.

Step by step solution

01

Identify the Leading Term

First, identify the leading term of the polynomial. In this case, it's \(11x^{4}\) since this term has the highest degree (power of x).
02

Identify the Leading Coefficient and Degree

Next, identify the leading coefficient and the degree of the polynomial. In this case, the leading coefficient (the number in front of the variable \(x\)) is 11 and the degree (the highest power of \(x\)) is 4.
03

Applying the Leading Coefficient Test

As our leading coefficient is positive and the degree is even, by the Leading Coefficient Test, when \(x\) goes to both positive infinity and negative infinity, \(f(x)\) will head towards positive infinity.

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