To analyze the function [tex]\( f(x) = x^3 + 2x^2 - 5x - 6 \)[/tex], we need to find the zeros of the function and describe its end behavior.
### Step 1: Finding the Zeros of the Function
The zeros of the function are the values of [tex]\( x \)[/tex] where [tex]\( f(x) = 0 \)[/tex]. After solving this equation:
[tex]\[ x^3 + 2x^2 - 5x - 6 = 0 \][/tex]
We find that the solutions (zeros) are:
[tex]\[ x = -3, \, x = -1, \, x = 2 \][/tex]
So, the zeros of the function on the coordinate plane are at [tex]\( (-3, 0) \)[/tex], [tex]\( (-1, 0) \)[/tex], and [tex]\( (2, 0) \)[/tex].
### Step 2: Determining the End Behavior
For a cubic function of the form [tex]\( f(x) = ax^3 + bx^2 + cx + d \)[/tex], the end behavior can be determined as follows:
- As [tex]\( x \to \infty \)[/tex], the function [tex]\( f(x) \to \infty \)[/tex] if the leading coefficient [tex]\( a \)[/tex] is positive or [tex]\( f(x) \to -\infty \)[/tex] if [tex]\( a \)[/tex] is negative.
- As [tex]\( x \to -\infty \)[/tex], the function [tex]\( f(x) \to -\infty \)[/tex] if the leading coefficient [tex]\( a \)[/tex] is positive or [tex]\( f(x) \to \infty \)[/tex] if [tex]\( a \)[/tex] is negative.
For our function [tex]\( f(x) = x^3 + 2x^2 - 5x - 6 \)[/tex]:
- The leading coefficient [tex]\( a \)[/tex] is 1 (positive), so as [tex]\( x \to \infty \)[/tex], [tex]\( f(x) \to \infty \)[/tex].
- As [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \to -\infty \)[/tex].
To confirm, we check the function's values at very large and very small [tex]\( x \)[/tex]:
- For [tex]\( x = 1,000,000 \)[/tex], the function's value [tex]\( f(1,000,000) \)[/tex] is [tex]\( 1,000,001,999,994,999,994 \)[/tex] indicating [tex]\( f(x) \to \infty \)[/tex].
- For [tex]\( x = -1,000,000 \)[/tex], the function's value [tex]\( f(-1,000,000) \)[/tex] is [tex]\( -999,997,999,995,000,006 \)[/tex] indicating [tex]\( f(x) \to -\infty \)[/tex].
### Conclusion
Summarizing our findings:
- The zeros of the function are at [tex]\( (-3, 0) \)[/tex], [tex]\( (-1, 0) \)[/tex], and [tex]\( (2, 0) \)[/tex].
- The end behavior of the function is:
- As [tex]\( x \to \infty \)[/tex], [tex]\( f(x) \to \infty \)[/tex].
- As [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \to -\infty \)[/tex].
