Question

question 3 of 5
select the correct answer from each drop-down menu.
consider the given function.
$f(x) = x^4 - 3x^3 + 4x^2 - 12x$
function $f$ has distinct real zeros and complex zeros.

Explanation:

Step1: Factor the function by grouping

Group terms and factor out common factors:

$$\begin{align*} f(x)&=x^4 - 3x^3 + 4x^2 - 12x\\ &=x^3(x - 3) + 4x(x - 3)\\ &=(x - 3)(x^3 + 4x) \end{align*}$$

Step2: Factor further

Factor out $x$ from the second term:

$$ f(x)=x(x - 3)(x^2 + 4) $$

Step3: Find real zeros

Set $f(x)=0$ and solve for real $x$:
$x=0$, $x-3=0 \implies x=3$. The term $x^2 + 4=0$ gives $x^2=-4$, which has no real solutions.

Step4: Find complex zeros

Solve $x^2 + 4=0$:

$$ x^2=-4 \implies x=\pm 2i $$

Count distinct real and complex zeros: 2 distinct real zeros, 2 complex zeros.

Answer:

Function $f$ has $\boldsymbol{2}$ distinct real zeros and $\boldsymbol{2}$ complex zeros.