Question

solve the polynomial equation by factoring and then using the zero - product principle.
$7x^{4}=1512x$
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 one side

Subtract \(1512x\) from both sides of the equation \(7x^4 = 1512x\) to get \(7x^4-1512x = 0\).

Step2: Factor out the greatest common factor

Factor out \(7x\) from the left - hand side: \(7x(x^3 - 216)=0\).

Step3: Recognize the difference of cubes

We know that \(a^3 - b^3=(a - b)(a^2+ab + b^2)\). For \(x^3-216\), we have \(a = x\) and \(b = 6\) (since \(6^3=216\)), so \(x^3 - 216=(x - 6)(x^2+6x + 36)\). Then the equation becomes \(7x(x - 6)(x^2+6x + 36)=0\).

Step4: Apply the zero - product principle

The zero - product principle states that if \(ab = 0\), then either \(a = 0\) or \(b = 0\).

• For \(7x=0\), we get \(x = 0\).
• For \(x - 6=0\), we get \(x = 6\).
• For \(x^2+6x + 36=0\), we use the quadratic formula \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\) where \(a = 1\), \(b = 6\), and \(c = 36\). The discriminant \(\Delta=b^2-4ac=6^2-4\times1\times36=36 - 144=- 108\). Then \(x=\frac{-6\pm\sqrt{-108}}{2}=\frac{-6\pm6\sqrt{3}i}{2}=-3\pm3\sqrt{3}i\).

Answer:

The solution set is \(\{0,6,-3 + 3\sqrt{3}i,-3 - 3\sqrt{3}i\}\)