Question

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

f(x)=78x^{2}-2x^{4}+84 - 6x^{3}+166x

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→ - ∞}f(x)=∞, lim_{x→∞}f(x)=∞

lim_{x→ - ∞}f(x)= - ∞, lim_{x→∞}f(x)=∞

lim_{x→ - ∞}f(x)=∞, lim_{x→∞}f(x)= - ∞

lim_{x→ - ∞}f(x)= - ∞, lim_{x→∞}f(x)= - ∞

Explanation:

Step1: Identify degree and leading - coefficient

For the polynomial $f(x)=78x^{2}-2x^{4}+84 - 6x^{3}+166x$, rewrite it in standard form $f(x)=-2x^{4}-6x^{3}+78x^{2}+166x + 84$. The degree of a polynomial is the highest power of the variable, so the degree $n = 4$ (even), and the leading - coefficient $a=-2$ (negative).

Step2: Determine end - behavior

When the degree $n$ of a polynomial is even and the leading coefficient $a<0$, as $x\to-\infty$, $f(x)=-2x^{4}-6x^{3}+78x^{2}+166x + 84\to-\infty$ because the term $-2x^{4}$ dominates (the term with the highest power). As $x\to\infty$, $f(x)=-2x^{4}-6x^{3}+78x^{2}+166x + 84\to-\infty$ since the leading term $-2x^{4}$ dominates.

Answer:

The degree of the polynomial is even, and the leading coefficient is negative. The pair of limits is $\lim_{x\to-\infty}f(x)=-\infty,\lim_{x\to\infty}f(x)=-\infty$