Question

question
factor to find all x-intercepts of the function.
$f(x) = 5x^5 + 20x^4 + 20x^3$
answer attempt 1 out of 2
additional solution no solution

Explanation:

Step1: Factor out the GCF

The greatest common factor (GCF) of \(5x^5\), \(20x^4\), and \(20x^3\) is \(5x^3\). So we factor that out:
\(f(x) = 5x^3(x^2 + 4x + 4)\)

Step2: Factor the quadratic

The quadratic \(x^2 + 4x + 4\) is a perfect square trinomial, which factors as \((x + 2)^2\). So now we have:
\(f(x) = 5x^3(x + 2)^2\)

Step3: Find the x - intercepts

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

• For \(5x^3=0\), we get \(x^3 = 0\), so \(x = 0\) (with multiplicity 3).
• For \((x + 2)^2=0\), we get \(x+2 = 0\), so \(x=-2\) (with multiplicity 2).

Answer:

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