Question

find all y-intercepts and x-intercepts of the graph of the function.
$f(x) = 3x^3 + 3x^2 - 18x$
if there is more than one answer, separate them with commas.
click on
one\ if applicable.

y-intercept(s):

x-intercept(s):

Explanation:

Step1: Find y-intercept

To find the y-intercept, set \( x = 0 \) in the function \( f(x)=3x^{3}+3x^{2}-18x \).
Substitute \( x = 0 \) into the function: \( f(0)=3(0)^{3}+3(0)^{2}-18(0)=0 \). So the y-intercept is \( 0 \).

Step2: Find x-intercepts

To find the x-intercepts, set \( f(x)=0 \), so we solve the equation \( 3x^{3}+3x^{2}-18x = 0 \).
First, factor out the greatest common factor, which is \( 3x \): \( 3x(x^{2}+x - 6)=0 \).
Then factor the quadratic expression \( x^{2}+x - 6 \). We need two numbers that multiply to \( - 6 \) and add to \( 1 \). Those numbers are \( 3 \) and \( - 2 \), so \( x^{2}+x - 6=(x + 3)(x - 2) \).
So the factored form of the equation is \( 3x(x + 3)(x - 2)=0 \).
Set each factor equal to zero:

• \( 3x=0\Rightarrow x = 0 \)
• \( x + 3=0\Rightarrow x=-3 \)
• \( x - 2=0\Rightarrow x = 2 \)

Answer:

y - intercept(s): \( 0 \)
x - intercept(s): \( -3, 0, 2 \)