Question

use factoring to solve the polynomial equation. check by substitution or by using a graphing utility and identifying x-intercepts.\\(3x^4 - 27x^2 = 0\\)\\(\\)find the solution set. select the correct choice below and, if necessary, fill in the answer box to complete your choice.\\(\\)\\(\bigcirc\\) a. the solution set is \\(\square\\)\\(\\) (use a comma to separate answers as needed. type each solution only once.)\\(\\)\\(\bigcirc\\) b. there is no solution.

Explanation:

Step1: Factor out the GCF

First, factor out the greatest common factor (GCF) from the polynomial \(3x^4 - 27x^2\). The GCF of \(3x^4\) and \(27x^2\) is \(3x^2\). So we have:
\(3x^2(x^2 - 9) = 0\)

Step2: Factor the difference of squares

Notice that \(x^2 - 9\) is a difference of squares, which can be factored as \((x - 3)(x + 3)\). So now the equation becomes:
\(3x^2(x - 3)(x + 3) = 0\)

Step3: Set each factor equal to zero and solve

Set each factor equal to zero and solve for \(x\):

• For \(3x^2 = 0\), divide both sides by 3: \(x^2 = 0\), so \(x = 0\) (with multiplicity 2, but we list it once in the solution set).
• For \(x - 3 = 0\), add 3 to both sides: \(x = 3\).
• For \(x + 3 = 0\), subtract 3 from both sides: \(x = -3\).

Answer:

A. The solution set is \(-3, 0, 3\)