Question

the function f(x) is defined below. what is the end behavior of f(x)? f(x)=2688 - 40x^4+4976x + 8x^5-504x^3+1832x^2 answer attempt 1 out of 2 the degree of the polynomial is , and the leading coefficient is . choose which pair of limits below odd 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 of polynomial

The degree of a polynomial is the highest power of the variable. For \(f(x)=2688 - 40x^{4}+4976x + 8x^{5}-504x^{3}+1832x^{2}\), the highest - power of \(x\) is \(5\), so the degree is \(5\) (an odd number).

Step2: Identify leading coefficient

The leading coefficient is the coefficient of the term with the highest power of the variable. For \(f(x)\), the coefficient of \(x^{5}\) is \(8\) (a positive number).

Step3: Determine end - behavior

For a polynomial \(y = a_nx^n+\cdots+a_0\) with \(n\) odd and \(a_n>0\), 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 \(5\) (odd), and the leading coefficient is \(8\). The pair of limits that represents the end - behavior is \(\lim_{x\to-\infty}f(x)=-\infty,\lim_{x\to\infty}f(x)=\infty\)