Question

question
the function ( f(x) ) is defined below. what is the end behavior of ( f(x) )?
f(x)=3 x^{5}-96 x^{2}-18 x^{3}-105 x-36+12 x^{4}
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
ightarrow-infty} f(x)=infty, quad lim _{x
ightarrow infty} f(x)=infty )
( lim _{x
ightarrow-infty} f(x)=-infty, quad lim _{x
ightarrow infty} f(x)=infty )
( lim _{x
ightarrow-infty} f(x)=infty, quad lim _{x
ightarrow infty} f(x)=-infty )
( lim _{x
ightarrow-infty} f(x)=-infty, quad lim _{x
ightarrow infty} f(x)=-infty )

Explanation:

Step1: Find the degree of the polynomial

The degree of a polynomial is the highest power of \(x\) in its expression. For \(f(x) = 3x^{5}-96x^{2}-18x^{3}-105x - 36+12x^{4}\), the powers of \(x\) are \(5\), \(2\), \(3\), \(1\), \(0\), \(4\). The highest power is \(n = 5\).

Step2: Find the leading coefficient

The leading coefficient is the coefficient of the term with the highest - degree. The term with the highest degree is \(3x^{5}\), so the leading coefficient \(a=3\).

Step3: Determine the end - behavior

For a polynomial \(y = ax^{n}+...\) (where \(a
eq0\) and \(n\) is a positive integer):

• If \(n\) is odd:
• When \(a>0\), as \(x

ightarrow\infty\), \(y = ax^{n}+...\approx ax^{n}\), and \(y
ightarrow\infty\) (since \(x^{n}>0\) for \(x>0\) and \(a = 3>0\)). As \(x
ightarrow-\infty\), \(y=ax^{n}+...\approx ax^{n}\), and \(x^{n}<0\) (because \(n = 5\) is odd) and \(a = 3>0\), so \(y
ightarrow-\infty\).

So the degree is \(5\), the leading coefficient is \(3\), and the end - behavior is \(\lim_{x
ightarrow-\infty}f(x)=-\infty,\lim_{x
ightarrow\infty}f(x)=\infty\).

Answer:

The degree of the polynomial is \(5\), and the leading coefficient is \(3\). The correct pair of limits is \(\lim_{x
ightarrow-\infty}f(x)=-\infty,\lim_{x
ightarrow\infty}f(x)=\infty\) (the second option).