To solve the problem, we need to find the zeros of the given function [tex]\( f(x) = x^3 + 2x^2 - 5x - 6 \)[/tex] and determine its end behavior.
### Step 1: Find the Zeros of the Function
The zeros of the function are the values of [tex]\( x \)[/tex] at which [tex]\( f(x) = 0 \)[/tex]. These are the points where the graph intersects the x-axis.
By solving the equation [tex]\( x^3 + 2x^2 - 5x - 6 = 0 \)[/tex], we find the following zeros:
[tex]\[ x = -3, -1, 2 \][/tex]
So, the zeros of the function [tex]\( f(x) \)[/tex] are at the points:
[tex]\[ (-3, 0), (-1, 0), (2, 0) \][/tex]
### Step 2: Determine the End Behavior of the Graph
The end behavior of a polynomial function is determined by the term with the highest degree. For [tex]\( f(x) = x^3 + 2x^2 - 5x - 6 \)[/tex], the highest degree term is [tex]\( x^3 \)[/tex].
- For large positive values of [tex]\( x \)[/tex]:
Since [tex]\( x^3 \)[/tex] grows very large positively, [tex]\( f(x) \)[/tex] also approaches infinity.
[tex]\[
\text{As } x \to \infty, \, f(x) \to \infty
\][/tex]
- For large negative values of [tex]\( x \)[/tex]:
Since [tex]\( x^3 \)[/tex] grows very large negatively, [tex]\( f(x) \)[/tex] also approaches negative infinity.
[tex]\[
\text{As } x \to -\infty, \, f(x) \to -\infty
\][/tex]
### Summary
- The zeros of the function [tex]\( f(x) = x^3 + 2x^2 - 5x - 6 \)[/tex] are located at [tex]\( x = -3, -1, \)[/tex] and [tex]\( 2 \)[/tex].
- The end behavior of the graph of [tex]\( f(x) \)[/tex] is:
[tex]\[
\begin{cases}
f(x) \to \infty & \text{as } x \to \infty \\
f(x) \to -\infty & \text{as } x \to -\infty
\end{cases}
\][/tex]
These findings indicate that the graph of [tex]\( f(x) \)[/tex] crosses the x-axis at the points [tex]\((-3, 0)\)[/tex], [tex]\((-1, 0)\)[/tex], and [tex]\((2, 0)\)[/tex] and extends towards positive infinity as [tex]\( x \to \infty \)[/tex] and negative infinity as [tex]\( x \to -\infty \)[/tex].
