Question

Solve for to determine if the
equation has one, none, or
infinite solutions.
(1 point)
x
3 (x + 2) − 6 = 3x
The equation has
infinite solutions.
The equation has one
solution: x = 3.
The equation has one
solution: x = −3.
The equation has no
solution.

Answer 1

To determine whether the equation \( x^3(x + 2) - 6 = 3x \) has one solution, no solution, or infinite solutions, we can first simplify and rearrange the equation.

Given equation:
\[
x^3(x + 2) - 6 = 3x
\]

Rearranging gives:
\[
x^3(x + 2) - 3x - 6 = 0
\]

Now let's expand \( x^3(x + 2) \):
\[
x^4 + 2x^3 - 3x - 6 = 0
\]

To find the solutions, we can analyze this polynomial equation. We can look for rational solutions using the Rational Root Theorem or look for specific values of \( x \) to see if they satisfy the equation.

Let’s try some values:

1. **Substituting \( x = 1 \):**
\[
1^4 + 2(1^3) - 3(1) - 6 = 1 + 2 - 3 - 6 = -6 \quad (\text{not a root})
\]

2. **Substituting \( x = -1 \):**
\[
(-1)^4 + 2(-1)^3 - 3(-1) - 6 = 1 - 2 + 3 - 6 = -4 \quad (\text{not a root})
\]

3. **Substituting \( x = 2 \):**
\[
2^4 + 2(2^3) - 3(2) - 6 = 16 + 16 - 6 - 6 = 20 \quad (\text{not a root})
\]

4. **Substituting \( x = -2 \):**
\[
(-2)^4 + 2(-2)^3 - 3(-2) - 6 = 16 - 16 + 6 - 6 = 0 \quad (\text{this is a root})
\]

Since \( x = -2 \) is a solution, we can factor out \( (x + 2) \) from the polynomial \( x^4 + 2x^3 - 3x - 6 \).

To factor the polynomial, we can perform synthetic division or polynomial long division using \( x + 2 \).

Doing the synthetic division:

\[
\begin{array}{r|rrrrr}
-2 & 1 & 2 & 0 & -3 & -6 \\
& & -2 & 0 & 0 & 6 \\
\hline
& 1 & 0 & 0 & -3 & 0 \\
\end{array}
\]

After division, we have:

\[
x^4 + 2x^3 - 3x - 6 = (x + 2)(x^3 - 3)
\]

Now, we can set the remaining factor to zero and solve:

\[
x^3 - 3 = 0 \implies x^3 = 3 \implies x = \sqrt[3]{3}
\]

Thus, the solutions to the original equation are:
1. \( x = -2 \)
2. \( x = \sqrt[3]{3} \)

Given that there are two distinct solutions, we conclude that:

**The equation has one solution, \( x = -2 \); and one more solution, \( x = \sqrt[3]{3} \), hence there are two solutions, but the provided choices suggest we cannot choose more than one solution per your context. Therefore, the statement about having infinite solutions is incorrect.**

To summarize, the answer is:
**The equation has two distinct solutions.**

If the answer must come from your given options, it may not align perfectly with the findings, but the accurate statement based on polynomial solutions is that it has two solutions: \( x = -2 \) and \( x = \sqrt[3]{3} \).