Answer:
Our y-intercept is (0, -36).
Our real zeros are: (-2√3, 0) and (2√3, 0)
And our imaginary zeros are: (i√3, 0) and (-i√3, 0)
Step-by-step explanation:
We have the function:
Let's identify the y- and x-intercepts.
Y-Intercept:
To determine the y-intercept, we will substitute 0 for x and solve for y. So:
Evaluate:
Therefore, our y-intercept is at (0, -36).
X-Intercepts:
To determine our x-intercepts, we need to substitute 0 for y and solve for x. So:
Now, we can factor. Before doing so, we can make the factored easier to see by converting this into quadratic form. Let's let u=x². Then:
Substitute:
Now, let's factor. We can use -12 and 3. So:
Now, we can substitute back u:
Zero Product Property:
Solve for x in each case:
Take the square root of both sides. Since we're taking an even root, we will need plus/minus. Therefore:
Therefore, our zeros (both real and imaginary) are: (2√3, 0), (-2√3, 0), (i√3, 0), and (-i√3, 0)