Question

solve the polynomial equation by factoring and then using the zero - product principle.
$x^{3}+2x^{2}=81x + 162$
rewrite the equation in factored form.
$\square=0$
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. simplify your answer.)
\\(\bigcirc\\) b. there is no solution.

Explanation:

Step1: Move all terms to left

To factor the polynomial, first move all terms to the left side of the equation. So we have \(x^{3}+2x^{2}-81x - 162=0\).

Step2: Group terms

Group the first two terms and the last two terms: \((x^{3}+2x^{2})+(-81x - 162)=0\).

Step3: Factor out GCF from each group

Factor out the greatest common factor (GCF) from each group. From the first group \(x^{3}+2x^{2}\), the GCF is \(x^{2}\), so we get \(x^{2}(x + 2)\). From the second group \(-81x-162\), the GCF is \(-81\), so we get \(-81(x + 2)\). Now the equation becomes \(x^{2}(x + 2)-81(x + 2)=0\).

Step4: Factor out common binomial

Notice that both terms have a common binomial factor of \((x + 2)\). Factor that out: \((x + 2)(x^{2}-81)=0\).

Step5: Factor the difference of squares

The term \(x^{2}-81\) is a difference of squares, which can be factored as \((x - 9)(x + 9)\) (since \(a^{2}-b^{2}=(a - b)(a + b)\) with \(a=x\) and \(b = 9\)). So the fully factored form is \((x + 2)(x - 9)(x + 9)=0\).

Step6: Apply zero - product principle

The zero - product principle states that if \(ab = 0\), then either \(a=0\) or \(b = 0\). So we set each factor equal to zero:

• \(x+2=0\), then \(x=-2\).
• \(x - 9=0\), then \(x = 9\).
• \(x+9=0\), then \(x=-9\).

Answer:

The factored form is \((x + 2)(x - 9)(x + 9)=0\) (for the first part of the question). The solution set is \(\{-9,-2,9\}\) (so for the solution set, the correct choice is A and the solution set is \(\{-9, - 2,9\}\)).