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\
\\( \oplus \\) additional solution \\( \circleddash \\) no solution\
\\( x = \\) input box submit ans

Explanation:

Step1: Factor out common term \(x\)

First, we can factor out an \(x\) from each term of the function \(f(x) = x^4 - 3x^3 + 4x^2 - 12x\). So we get:
\(f(x)=x(x^{3}-3x^{2}+4x - 12)\)

Step2: Group terms in the cubic polynomial

Now, we group the terms in the cubic polynomial \(x^{3}-3x^{2}+4x - 12\). We group the first two terms and the last two terms:
\(x^{3}-3x^{2}+4x - 12=(x^{3}-3x^{2})+(4x - 12)\)

Step3: Factor out common factors from each group

Factor out \(x^{2}\) from the first group and \(4\) from the second group:
\((x^{3}-3x^{2})+(4x - 12)=x^{2}(x - 3)+4(x - 3)\)

Step4: Factor out the common binomial factor

Now, we can see that both terms have a common factor of \((x - 3)\). So we factor that out:
\(x^{2}(x - 3)+4(x - 3)=(x^{2}+4)(x - 3)\)

Step5: Combine all factors

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

Step6: Find the x-intercepts

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

• For \(x = 0\), the equation is satisfied.
• For \(x^{2}+4=0\), we have \(x^{2}=- 4\), and the solutions are \(x=\pm2i\) (these are complex solutions).
• For \(x - 3=0\), we get \(x = 3\).

The x - intercepts occur at the real solutions of the equation \(f(x)=0\). So the real x - intercepts are \(x = 0\) and \(x=3\).

Answer:

The x - intercepts of the function are \(x = 0\) and \(x = 3\). If we consider the boxed answer (for one of them, but since there are two real x - intercepts), if we take \(x = 0\) (or \(x = 3\)), for example, the answer is \(\boxed{0}\) (or \(\boxed{3}\)).