Question

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

f(x)=52x^{2}-4x^{3}-196 - 140x

answer attempt 1 out of 2

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

choose which pair of limits below odd ents 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 of polynomial

The highest - power of \(x\) in \(f(x)=52x^{2}-4x^{3}-196 - 140x\) is 3, so the degree is 3 (odd).

Step2: Identify leading coefficient

The term with the highest - power of \(x\) is \(-4x^{3}\), so the leading coefficient is \(- 4\) (negative).

Step3: Determine end - behavior using rules

For a polynomial \(y = a_nx^n+\cdots+a_0\) with odd degree \(n\) and negative leading coefficient \(a_n\), 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 negative. The pair of limits that represents the end - behavior is \(\lim_{x\to-\infty}f(x)=\infty\), \(\lim_{x\to\infty}f(x)=-\infty\)