Question

finding x-intercepts in unfactored form
this is the only question in this section.
question
factor to find all x-intercepts of the function.
$f(x) = 3x^5 - 4x^4 - 20x^3$
answer

Explanation:

Step1: Factor out the GCF

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

Step2: Factor the quadratic

Now we factor the quadratic \(3x^2 - 4x - 20\). We need two numbers that multiply to \(3\times(-20)=-60\) and add to \(-4\). Those numbers are \(-10\) and \(6\).
So we rewrite the middle term:
\(3x^2+6x - 10x - 20\)
Group the terms:
\((3x^2 + 6x)+(-10x - 20)\)
Factor out the GCF from each group:
\(3x(x + 2)-10(x + 2)\)
Now we can factor out \((x + 2)\):
\((3x - 10)(x + 2)\)
So the fully factored form of \(f(x)\) is:
\(f(x)=x^3(3x - 10)(x + 2)\)

Step3: Find the x - intercepts

To find the x - intercepts, we set \(f(x)=0\):
\(x^3(3x - 10)(x + 2)=0\)
Using the zero - product property, if \(ab = 0\), then either \(a = 0\) or \(b = 0\).

• For \(x^3=0\), we have \(x = 0\) (with multiplicity 3).
• For \(3x - 10=0\), we solve for \(x\): \(3x=10\), so \(x=\frac{10}{3}\).
• For \(x + 2=0\), we solve for \(x\): \(x=-2\)

Answer:

The x - intercepts are \(x = 0\), \(x=-2\), and \(x=\frac{10}{3}\)