Question

find all zeros of $f(x) = x^4 - 6x^3 + 2x^2 + 24x - 24$. enter the zeros separated by commas. enter exact values, not decimal approximations.

Explanation:

Step1: Try rational root theorem

Possible rational roots: $\pm1, \pm2, \pm3, \pm4, \pm6, \pm8, \pm12, \pm24$.
Test $x=2$: $f(2)=2^4-6(2)^3+2(2)^2+24(2)-24=16-48+8+48-24=0$.
So $(x-2)$ is a factor.

Step2: Polynomial division

Divide $x^4-6x^3+2x^2+24x-24$ by $(x-2)$:
$$\frac{x^4-6x^3+2x^2+24x-24}{x-2}=x^3-4x^2-6x+12$$

Step3: Factor cubic polynomial

Group terms: $(x^3-4x^2)+(-6x+12)=x^2(x-4)-6(x-4)=(x^2-6)(x-4)$

Step4: Find all roots

Set each factor to 0:

1. $x-2=0 \implies x=2$
2. $x-4=0 \implies x=4$
3. $x^2-6=0 \implies x=\pm\sqrt{6}$

Answer:

$2, 4, \sqrt{6}, -\sqrt{6}$