**The volume of a rectangular prism is defined by the function, *V*(*x*) = *x*3 + 5*x*2 - 2*x* - 24. Use the zeros of the function to represent the volume of the prism, *V*(*x*), as the product of its linear factors.**

*V*( *x*) = x (x - 2) (x +3)

*V*( *x*) = (x - 4) (x + 2) (x - 3)

*V*( *x*) = (x + 4) (x - 3) (x + 3)

*V*( *x*) = (x + 4) (x - 2) (x + 3)

To solve for the zeros of the function \( V(x) = x^3 + 5x^2 - 2x - 24 \) and represent the volume as the product of its linear factors, let's follow these steps:

### Step 1: Find the Zeros of the Polynomial
We are given \( V(x) = x^3 + 5x^2 - 2x - 24 \). To find its factors, we need to determine the zeros of the polynomial.

1. Start by using the **Rational Root Theorem** to test possible rational roots. The potential rational roots are the factors of the constant term (-24) divided by the factors of the leading coefficient (1). The possible rational roots are:
   \[
   \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24
   \]
   
2. Test these values by substituting them into the polynomial until we find a zero.

   - For \( x = -4 \):
     \[
     V(-4) = (-4)^3 + 5(-4)^2 - 2(-4) - 24 = -64 + 80 + 8 - 24 = 0
     \]
     So, \( x = -4 \) is a root.

### Step 2: Use Synthetic Division
Now that we know \( x = -4 \) is a root, we can use synthetic division to divide \( V(x) \) by \( x + 4 \).

\[
\begin{array}{r|rrrr}
-4 & 1 & 5 & -2 & -24 \\
   &   & -4 & -4 & 24 \\
\hline
   & 1 & 1  & -6 & 0 \\
\end{array}
\]
The quotient is \( x^2 + x - 6 \).

### Step 3: Factor the Quotient
Now, we factor \( x^2 + x - 6 \). This is a simple quadratic that can be factored as:
\[
x^2 + x - 6 = (x - 2)(x + 3)
\]

### Step 4: Write the Complete Factorization
Now that we have factored the cubic polynomial, the complete factorization of \( V(x) \) is:
\[
V(x) = (x + 4)(x - 2)(x + 3)
\]

Thus, the correct factorization of the volume function is:

\[
\boxed{V(x) = (x + 4)(x - 2)(x + 3)}
\]

### Answer:
The correct option is:
\[
V(x) = (x + 4)(x - 2)(x + 3)
\]

Would you like more details or have any questions?

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Here are 5 related questions to deepen your understanding:
1. How does the Rational Root Theorem help in finding the factors of a polynomial?
2. What other methods could be used to factor cubic polynomials like this one?
3. How would you graph the function \( V(x) = x^3 + 5x^2 - 2x - 24 \) based on its factored form?
4. How can we verify the factorization by expanding \( (x + 4)(x - 2)(x + 3) \)?
5. What is the significance of the zeros of \( V(x) \) in relation to the dimensions of the rectangular prism?

**Tip**: Always verify your factorization by expanding the linear factors to ensure they match the original polynomial.

The volume of a rectangular prism is defined by the function, V(x) = x^3 + 5x^2 - 2x - 24. Use the zeros of the function to represent the volume of the prism, V(x), as the product of its linear factors. V(x) = x (x - 2) (x +3), V(x) = (x - 4) (x + 2) (x - 3), V(x) = (x + 4) (x - 3) (x + 3), V(x) = (x + 4) (x - 2) (x + 3).

Learn how to factor a cubic polynomial V(x) = x^3 + 5x^2 - 2x - 24 to find the volume of a rectangular prism. This step-by-step solution covers polynomial factorization, the Rational Root Theorem, and finding linear factors. Ideal for students in Grades 10-12.

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