Question

question
factor to find all x-intercepts of the function.
$f(x) = x^4 - 3x^3 + 4x^2 - 12x$
answer
attempt 1 out of 2

Explanation:

Step1: Factor out the GCF

The greatest common factor (GCF) of the terms \(x^4\), \(-3x^3\), \(4x^2\), and \(-12x\) is \(x\). So we factor out \(x\):
\(f(x)=x(x^3 - 3x^2 + 4x - 12)\)

Step2: Group the cubic polynomial

Group the terms in the cubic polynomial \(x^3 - 3x^2 + 4x - 12\) as \((x^3 - 3x^2)+(4x - 12)\)

Step3: Factor each group

Factor \(x^2\) from the first group and \(4\) from the second group:
\(x^2(x - 3)+4(x - 3)\)

Step4: Factor out the common binomial

Now we can factor out \((x - 3)\) from the expression:
\((x - 3)(x^2 + 4)\)

Step5: Combine all factors

Putting it all together, the factored form of \(f(x)\) is:
\(f(x)=x(x - 3)(x^2 + 4)\)

Step6: Find the x - intercepts

To find the x - intercepts, we set \(f(x)=0\). So:
\(x(x - 3)(x^2 + 4)=0\)
Using the zero - product property, we set each factor equal to zero:

• For \(x = 0\), the equation is satisfied.
• For \(x-3=0\), we get \(x = 3\).
• For \(x^2+4=0\), we have \(x^2=-4\), and the solutions for this are \(x=\pm2i\) (which are not real numbers). Since we are looking for x - intercepts (real - valued), we consider the real solutions from the factors.

Answer:

The x - intercepts of the function \(f(x)=x^4 - 3x^3 + 4x^2 - 12x\) are \(x = 0\) and \(x = 3\) (if we consider real x - intercepts). If we list all real x - intercepts, the answer is \(x = 0,3\)