Question

question
factor to find all x-intercepts of the function.
$f(x) = 5x^3 + 5x^2 - 30x$
answer attempt 1 out of 2
+ additional solution - no solution
$x = \boxed{}$ submit ans

Explanation:

Step1: Factor out the GCF

The greatest common factor (GCF) of \(5x^{3}\), \(5x^{2}\), and \(-30x\) is \(5x\). So we factor that out:
\(f(x)=5x(x^{2}+x - 6)\)

Step2: Factor the quadratic

Now we factor the quadratic \(x^{2}+x - 6\). We need two numbers that multiply to \(-6\) and add to \(1\). Those numbers are \(3\) and \(-2\). So:
\(x^{2}+x - 6=(x + 3)(x - 2)\)

Step3: Find the x - intercepts

To find the x - intercepts, we set \(f(x)=0\). So:
\(5x(x + 3)(x - 2)=0\)
Using the zero - product property (if \(ab = 0\), then either \(a = 0\) or \(b = 0\)), we have:

• If \(5x=0\), then \(x = 0\).
• If \(x + 3=0\), then \(x=-3\).
• If \(x - 2=0\), then \(x = 2\).

Answer:

\(x=-3,0,2\)