Question

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

f(x)=5x^2 - x^3+22x - 56

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

The degree of a polynomial $a_nx^n + a_{n - 1}x^{n-1}+\cdots+a_0$ is the highest - power of $x$. For $f(x)=5x^2 - x^3+22x - 56$, the highest - power of $x$ is $n = 3$ (odd), and the leading coefficient (the coefficient of the term with the highest power of $x$) is $- 1$.

Step2: Determine end - behavior

For a polynomial $y = a_nx^n+\cdots+a_0$ with odd degree $n$ and negative leading coefficient $a_n<0$, as $x\to-\infty$, $y\to\infty$ and as $x\to\infty$, $y\to-\infty$. That is, $\lim_{x\to-\infty}f(x)=\infty$ and $\lim_{x\to\infty}f(x)=-\infty$.

Answer:

The degree of the polynomial is odd, and the leading coefficient is $-1$. The pair of limits that represents the end - behavior is $\lim_{x\to-\infty}f(x)=\infty,\lim_{x\to\infty}f(x)=-\infty$