Question

solve the polynomial equation by factoring and then using the zero - product principle
$4x^{4}=256x$

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

$4x^4 - 256x = 0$

Step2: Factor out common term $4x$

$4x(x^3 - 64) = 0$

Step3: Factor the difference of cubes

$x^3 - 64 = (x - 4)(x^2 + 4x + 16)$, so:
$4x(x - 4)(x^2 + 4x + 16) = 0$

Step4: Apply zero-product principle

Set each factor equal to 0:

1. $4x = 0 \implies x = 0$
2. $x - 4 = 0 \implies x = 4$
3. Solve $x^2 + 4x + 16 = 0$ using quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a=1, b=4, c=16$:

$\Delta = 4^2 - 4(1)(16) = 16 - 64 = -48$
$x = \frac{-4 \pm \sqrt{-48}}{2} = \frac{-4 \pm 4i\sqrt{3}}{2} = -2 \pm 2i\sqrt{3}$

Answer:

A. The solution set is $\{0, 4, -2+2i\sqrt{3}, -2-2i\sqrt{3}\}$