Consider $f f(x)=3 x^{4}+6 x^{3}$
a. What is the end behavior?
b. Write the function in factored form.
\[
f(x)=3 x^{3}(x+2)
\]
c. Find the zeros, including multiplicity.
Answer
Final Answer: The end behavior of the function \(f(x)=3 x^{4}+6 x^{3}\) is that as \(x\) approaches positive or negative infinity, \(f(x)\) approaches positive infinity. In other words, the ends of the graph are both up. This is represented as \(\lim_{{x \to \infty}} f(x) = \infty\) and \(\lim_{{x \to -\infty}} f(x) = \infty\).
Step 1 ：The end behavior of a function refers to the behavior of the function as x approaches positive infinity and negative infinity. For polynomial functions, the end behavior is determined by the degree and the leading coefficient. If the degree of the polynomial is even, then the ends of the graph are either both up or both down. If the degree is odd, then one end of the graph is up and the other is down. The leading coefficient determines whether the graph is up or down. If the leading coefficient is positive, the right end of the graph is up. If it is negative, the right end of the graph is down.
Step 2 ：In this case, the degree of the polynomial is 4, which is even, and the leading coefficient is 3, which is positive. Therefore, the ends of the graph are both up.
Step 3 ：The limits of the function as x approaches positive and negative infinity are both positive infinity. This confirms our earlier thought that the ends of the graph are both up.
Step 4 ：Final Answer: The end behavior of the function \(f(x)=3 x^{4}+6 x^{3}\) is that as \(x\) approaches positive or negative infinity, \(f(x)\) approaches positive infinity. In other words, the ends of the graph are both up. This is represented as \(\lim_{{x \to \infty}} f(x) = \infty\) and \(\lim_{{x \to -\infty}} f(x) = \infty\).
