Question

factor to find all x-intercepts of the function.\\( f(x) = -x^3 - 3x^2 + 10x \\)\
answer attempt 1 out of 2\
\\( oplus \\) additional solution \\( ominus \\) no solution\
\\( x = \\) input box submit answer

Explanation:

Step1: Factor out -x

First, we factor out \(-x\) from the function \(f(x)=-x^{3}-3x^{2}+10x\).
\(f(x)=-x(x^{2}+3x - 10)\)

Step2: Factor the quadratic

Next, we factor the quadratic \(x^{2}+3x - 10\). We need two numbers that multiply to \(- 10\) and add to \(3\). The numbers are \(5\) and \(-2\). So, \(x^{2}+3x - 10=(x + 5)(x-2)\)

Step3: Set f(x) = 0 and solve for x

Now we set \(f(x) = 0\):
\(-x(x + 5)(x - 2)=0\)
Using the zero - product property, if \(ab = 0\), then either \(a = 0\), \(b=0\) or both.

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

Answer:

The x - intercepts are \(x = 0\), \(x=-5\) and \(x = 2\)