Question

the function f(x) is defined below. what is the end - behavior of f(x)?

f(x)=1152x^{2}-3072 + 8x^{5}+152x^{3}-96x^{4}-2176x

answer attempt 1 out of 2

the degree of the polynomial is, and the leading coefficient is

choose which pair of limits below represents the end - behavior:

lim_{x\to - \infty}f(x)=\infty, lim_{x\to\infty}f(x)=\infty

lim_{x\to - \infty}f(x)=-\infty, lim_{x\to\infty}f(x)=\infty

lim_{x\to - \infty}f(x)=\infty, lim_{x\to\infty}f(x)=-\infty

lim_{x\to - \infty}f(x)=-\infty, lim_{x\to\infty}f(x)=-\infty

Explanation:

Step1: Identify degree and leading - coefficient

For the polynomial $f(x)=8x^{5}-96x^{4}+152x^{3}+1152x^{2}-2176x - 3072$, the degree is the highest power of $x$. Here, the degree $n = 5$ (odd), and the leading - coefficient $a = 8$ (positive).

Step2: Determine end - behavior using rules

For a polynomial $y = a_nx^n+\cdots+a_0$ with odd degree $n$ and positive leading - coefficient $a_n$:
As $x\to-\infty$, $y\to-\infty$ because when $x$ is a large negative number and $n$ is odd, $x^n$ is negative, and since $a_n>0$, $a_nx^n$ is negative.
As $x\to\infty$, $y\to\infty$ because when $x$ is a large positive number and $n$ is odd, $x^n$ is positive, and since $a_n>0$, $a_nx^n$ is positive. So, $\lim_{x\to-\infty}f(x)=-\infty$ and $\lim_{x\to\infty}f(x)=\infty$.

Answer:

The degree of the polynomial is odd, and the leading coefficient is 8. The pair of limits that represents the end - behavior is $\lim_{x\to-\infty}f(x)=-\infty$, $\lim_{x\to\infty}f(x)=\infty$