Question

solve the polynomial equation by factoring and then using the zero - product principle
$5x^{4}=1080x$

find the the solution set. select the correct choice below and, if necessary, fill in the answer box to complete your choice

a. the solution set is { }
(use a comma to separate answers as needed. simplify your answer. type your answer in the form $a + bi$)

b. there is no solution

Explanation:

Step1: Move all terms to left side

$5x^4 - 1080x = 0$

Step2: Factor out common term

$5x(x^3 - 216) = 0$

Step3: Factor the difference of cubes

Recall $a^3-b^3=(a-b)(a^2+ab+b^2)$. Here $a=x$, $b=6$ since $6^3=216$.
$5x(x - 6)(x^2 + 6x + 36) = 0$

Step4: Apply zero-product principle

Set each factor equal to 0:

1. $5x=0$
2. $x-6=0$
3. $x^2+6x+36=0$

Step5: Solve linear factors

For $5x=0$: $x=0$
For $x-6=0$: $x=6$

Step6: Solve quadratic factor

Use quadratic formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ for $ax^2+bx+c=0$. Here $a=1$, $b=6$, $c=36$.

$$\begin{align*} x&=\frac{-6\pm\sqrt{6^2-4(1)(36)}}{2(1)}\\ &=\frac{-6\pm\sqrt{36-144}}{2}\\ &=\frac{-6\pm\sqrt{-108}}{2}\\ &=\frac{-6\pm6\sqrt{3}i}{2}\\ &=-3\pm3\sqrt{3}i \end{align*}$$

Answer:

A. The solution set is $\{0, 6, -3+3\sqrt{3}i, -3-3\sqrt{3}i\}$