Question

Solve the polynomial equation by factoring and then using the​ zero-product principle.
x cubed plus 2 x squared equals 81 x plus 162
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Part 1
Find the solution set. Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.

Answer 1

To solve the given polynomial equation \( x^3 + 2x^2 = 81x + 162 \), we will first rearrange all terms to one side:

\[
x^3 + 2x^2 - 81x - 162 = 0
\]

Next, we will try to factor the equation. To do that, we can use the Rational Root Theorem to test possible rational roots, which in this case could be factors of -162. After testing some possible rational roots, we find that \( x = 9 \) is a root.

Let's perform synthetic division of the polynomial \( x^3 + 2x^2 - 81x - 162 \) by \( x - 9 \):

\[
\begin{array}{r|rrrr}
9 & 1 & 2 & -81 & -162 \\
& & 9 & 99 & 162 \\
\hline
& 1 & 11 & 18 & 0 \\
\end{array}
\]

So, the polynomial can be factored as:

\[
(x - 9)(x^2 + 11x + 18) = 0
\]

Next, we need to factor \( x^2 + 11x + 18 \). This can be factored into:

\[
(x + 2)(x + 9) = 0
\]

Now we can express the original polynomial in factored form:

\[
(x - 9)(x + 2)(x + 9) = 0
\]

Using the zero-product principle, we set each factor equal to zero:

1. \( x - 9 = 0 \) → \( x = 9 \)
2. \( x + 2 = 0 \) → \( x = -2 \)
3. \( x + 9 = 0 \) → \( x = -9 \)

Thus, the solution set is:

\[
\{9, -2, -9\}
\]

In conclusion, the solution to the polynomial equation is:

\(\{ 9, -2, -9 \}\)