8. An artist creates
a carving from a
rectangular block
of soapstone whose
volume, V, in cubic
metres, can be modelled by
V(x)  6x3 + 25x2  +2x-  8. Determine
possible dimensions of the block, in metres,
in terms of binomials of x

To determine the possible dimensions of the block of soapstone in terms of binomials of \( x \), we need to factor the volume polynomial \( V(x) = 6x^3 + 25x^2 + 2x - 8 \). The goal is to express the volume as a product of three binomials, each representing one dimension of the block.

### Step-by-Step Solution:

1. **Identify and apply the Rational Root Theorem:**
   The Rational Root Theorem helps us find possible rational roots of the polynomial. Possible rational roots are the factors of the constant term (-8) divided by the factors of the leading coefficient (6). The possible rational roots are:
   \[
   \pm 1, \pm 2, \pm 4, \pm 8, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{4}{3}, \pm \frac{8}{3}, \pm \frac{1}{6}
   \]

2. **Test possible rational roots:**
   Using synthetic division or direct substitution, test these possible roots to see if they satisfy the polynomial equation \( 6x^3 + 25x^2 + 2x - 8 = 0 \).

3. **Synthetic division:**
   Let’s test \( x = 1 \):

   \[
   \begin{array}{r|rrrr}
   1 & 6 & 25 & 2 & -8 \\
     &   & 6 & 31 & 33 \\
   \hline
     & 6 & 31 & 33 & 25 \\
   \end{array}
   \]

   Since the remainder is not zero, \( x = 1 \) is not a root. Continue this process for other possible roots.

   Testing \( x = -2 \):

   \[
   \begin{array}{r|rrrr}
   -2 & 6 & 25 & 2 & -8 \\
      &   & -12 & -26 & 48 \\
   \hline
      & 6 & 13 & -24 & 40 \\
   \end{array}
   \]

   Testing \( x = \frac{1}{2} \):

   \[
   \begin{array}{r|rrrr}
   \frac{1}{2} & 6 & 25 & 2 & -8 \\
               &   & 3 & 14 & 8 \\
   \hline
               & 6 & 28 & 16 & 0 \\
   \end{array}
   \]

   The remainder is 0, indicating \( x = \frac{1}{2} \) is a root.

4. **Factor the polynomial:**
   Using synthetic division to divide \( 6x^3 + 25x^2 + 2x - 8 \) by \( x - \frac{1}{2} \):

   \[
   \frac{6x^3 + 25x^2 + 2x - 8}{x - \frac{1}{2}} = 6x^2 + 28x + 16
   \]

   Now factor \( 6x^2 + 28x + 16 \):

   \[
   6x^2 + 28x + 16 = 2(3x^2 + 14x + 8)
   \]

   We can factor \( 3x^2 + 14x + 8 \) using the quadratic formula or factoring by grouping:

   \[
   3x^2 + 14x + 8 = (3x + 2)(x + 4)
   \]

5. **Combine all factors:**
   So, the volume polynomial \( 6x^3 + 25x^2 + 2x - 8 \) factors to:

   \[
   V(x) = (x - \frac{1}{2})(3x + 2)(x + 4)
   \]

### Conclusion:

The possible dimensions of the block, in metres, in terms of binomials of \( x \), are:
\[
\boxed{\left(x - \frac{1}{2}\right), (3x + 2), (x + 4)}
\]

Would you like any details or have any questions? Here are some possible follow-up questions:

1. Can you explain synthetic division in more detail?
2. How can we verify that the factorization is correct?
3. What are some applications of polynomial factorization in real life?
4. Could you show how to graph this polynomial function?
5. How does the Rational Root Theorem work exactly?

**Tip:** When factoring polynomials, always check your work by multiplying the factors to ensure you get the original polynomial.

Explore how to factor the cubic polynomial 6x^3 + 25x^2 + 2x - 8 into binomials using synthetic division and the Rational Root Theorem. This problem covers advanced algebraic concepts suitable for high school students.

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