Certainly! Let's solve the given problem step-by-step:
Given the function:
[tex]\[
y = 8 x^3 + \frac{6 x^4}{x^{-6}} - 7 x^5 + 1
\][/tex]
### Step (a): Simplify the function to identify the dominant term
First, we need to simplify the expression [tex]\(\frac{6 x^4}{x^{-6}}\)[/tex]:
[tex]\[
\frac{6 x^4}{x^{-6}} = 6 x^4 \cdot x^6 = 6 x^{4+6} = 6 x^{10}
\][/tex]
Now, substitute this back into the original function:
[tex]\[
y = 8 x^3 + 6 x^{10} - 7 x^5 + 1
\][/tex]
To determine the power function that the polynomial resembles, we look for the term with the highest degree:
[tex]\[
6 x^{10}
\][/tex]
Therefore, the power function that this polynomial resembles is:
[tex]\[
6 x^{10}
\][/tex]
### Step (b): Describe the long-run behavior of the polynomial
To describe the long-run behavior, we need to analyze the behavior of the polynomial as [tex]\(x\)[/tex] approaches positive and negative infinity.
#### As [tex]\( x \rightarrow \infty \)[/tex]:
The highest degree term in the polynomial is [tex]\(6 x^{10}\)[/tex]. As [tex]\( x \)[/tex] becomes very large, this term will dominate all the others. Since [tex]\(6 x^{10}\)[/tex] grows without bound as [tex]\( x \rightarrow \infty \)[/tex], we have:
[tex]\[
y \rightarrow \infty \text{ as } x \rightarrow \infty
\][/tex]
#### As [tex]\( x \rightarrow -\infty \)[/tex]:
Similarly, when [tex]\(x\)[/tex] becomes very large in the negative direction, the term [tex]\(6 x^{10}\)[/tex] also dominates all the others. Notably, since [tex]\(10\)[/tex] is an even power, [tex]\( x^{10} \)[/tex] will be positive even when [tex]\( x \)[/tex] is negative. Thus:
[tex]\[
y \rightarrow \infty \text{ as } x \rightarrow -\infty
\][/tex]
Therefore, the long-run behavior of the polynomial is:
[tex]\[
y \rightarrow \infty \text{ as } x \rightarrow \infty
\][/tex]
[tex]\[
y \rightarrow \infty \text{ as } x \rightarrow -\infty
\][/tex]
### Final Answers:
(a) The power function the polynomial resembles is:
[tex]\[
6 x^{10}
\][/tex]
(b) The long-run behavior is:
[tex]\[
y \rightarrow \infty \text{ as } x \rightarrow \infty
\][/tex]
[tex]\[
y \rightarrow \infty \text{ as } x \rightarrow -\infty
\][/tex]
