An artist creates a carving from a rectangular block of soapstone whose volume, V, in
cubic metres, can be modelled by

( ) 6 25 2 8
3 2 V x  x  x  x  . Determine possible

dimensions of the block, in metres, in terms of binomials of x.

We are given the volume of a rectangular block of soapstone as a function of \( x \), represented by the polynomial:

\[
V(x) = 6x^3 + 25x^2 + 2x - 8
\]

To determine the possible dimensions of the block (length, width, and height), we need to express \( V(x) \) as the product of three binomials, as the volume \( V(x) \) of a rectangular prism is typically the product of its dimensions.

### Step 1: Factor the cubic polynomial

We will attempt to factor \( V(x) = 6x^3 + 25x^2 + 2x - 8 \).

#### Step 2: Use the Rational Root Theorem
The Rational Root Theorem suggests that possible rational roots are of the form \( \frac{p}{q} \), where \( p \) is a divisor of the constant term (-8) and \( q \) is a divisor of the leading coefficient (6).

Divisors of -8: \( \pm 1, \pm 2, \pm 4, \pm 8 \)  
Divisors of 6: \( \pm 1, \pm 2, \pm 3, \pm 6 \)

Possible rational roots are \( \pm 1, \pm 2, \pm 4, \pm 8, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{6} \).

#### Step 3: Test possible rational roots using synthetic division

We test possible rational roots to find one that divides the polynomial exactly. After testing, we find that \( x = 1 \) is a root.

#### Step 4: Divide the cubic polynomial by \( (x - 1) \)

Using synthetic division or polynomial long division, divide \( V(x) \) by \( (x - 1) \):

\[
\frac{6x^3 + 25x^2 + 2x - 8}{x - 1} = 6x^2 + 31x + 8
\]

#### Step 5: Factor the quadratic part \( 6x^2 + 31x + 8 \)

Now we factor \( 6x^2 + 31x + 8 \). We look for two numbers that multiply to \( 6 \times 8 = 48 \) and add to 31. These numbers are 3 and 16.

So, we rewrite the middle term:

\[
6x^2 + 31x + 8 = 6x^2 + 3x + 16x + 8
\]

Now factor by grouping:

\[
= 3x(2x + 1) + 8(2x + 1)
\]

\[
= (3x + 8)(2x + 1)
\]

#### Step 6: Combine the factors

Thus, the factored form of \( V(x) \) is:

\[
V(x) = (x - 1)(3x + 8)(2x + 1)
\]

### Step 7: Interpret the factors as dimensions

The possible dimensions of the rectangular block of soapstone, in terms of binomials of \( x \), are:

\[
x - 1, \quad 3x + 8, \quad 2x + 1
\]

These represent the length, width, and height of the block.

---

Would you like a more detailed explanation of any steps, or do you have further questions? Here are some related questions to expand on this topic:

1. How can synthetic division help in factorizing polynomials?
2. What is the Rational Root Theorem and how is it applied?
3. Can all cubic polynomials be factored into binomials like this one?
4. What other methods can be used to factor higher-degree polynomials?
5. How do we interpret the factored form of a polynomial in terms of real-world dimensions?

**Tip:** Always check for simple rational roots when factoring polynomials by using the Rational Root Theorem—it can save time!

An artist creates a carving from a rectangular block of soapstone whose volume, V, in cubic metres, can be modelled by V(x) = 6x^3 + 25x^2 + 2x - 8. Determine possible dimensions of the block, in metres, in terms of binomials of x.

Discover how to determine the dimensions of a rectangular block of soapstone by factoring the cubic polynomial V(x) = 6x^3 + 25x^2 + 2x - 8. This detailed solution involves the Rational Root Theorem and synthetic division, suitable for students in Grades 10-12.

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