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Chapter 3: Problem 21

Use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. $$f(x)=5 x^{4}+7 x^{2}-x+9$$

Short Answer

Expert verified
The end behavior of the function \(f(x)=5 x^{4}+7 x^{2}-x+9\) is: As \(x\) approaches \(\pm\infty\), \(f(x)\) approaches \(+\infty\).

Step by step solution

01

Identifying the leading term

Identify the leading term of the polynomial function. This would be the term with the highest power of \(x\). In this case, the leading term is \(5x^{4}\).
02

Identifying the degree and leading coefficient

Determine the degree and the leading coefficient. The degree is the highest power of \(x\) in the leading term, which is 4, and the leading coefficient is the coefficient of the leading term, which is 5.
03

Applying the Leading Coefficient Test

Apply the Leading Coefficient Test considering two factors: 1) The degree, which is even, and 2) The leading coefficient, which is positive. According to the test, since the degree is even and the leading coefficient is positive, the function's end behavior can be described as: As \(x\) goes to positive infinity (\(+\infty\)), \(f(x)\) goes to positive infinity (\(+\infty\)). And as \(x\) goes to negative infinity (\(-\infty\)), \(f(x)\) also goes to positive infinity (\(+\infty\)).

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