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Let's start by tackling the first question from Exercise 4.6:

### Question 1a: 
"Show that \( x - 2 \) is a factor of \( 3x^3 - 14x^2 + 32 \)."

#### Solution Strategy:
To determine if \( x - 2 \) is a factor of the polynomial, we can apply synthetic division or use the factor theorem.

**Factor Theorem**:
- If \( x - c \) is a factor of a polynomial, then substituting \( x = c \) into the polynomial should give zero.

Let \( f(x) = 3x^3 - 14x^2 + 32 \). Substitute \( x = 2 \) into \( f(x) \):

\[
f(2) = 3(2)^3 - 14(2)^2 + 32 = 3(8) - 14(4) + 32 = 24 - 56 + 32 = 0.
\]

Since \( f(2) = 0 \), by the factor theorem, \( x - 2 \) is a factor of \( 3x^3 - 14x^2 + 32 \).

### Question 1b:
"Factorise \( 3x^3 - 14x^2 + 32 \) completely."

#### Solution Strategy:
We already know that \( x - 2 \) is a factor. Now, we can perform synthetic division or polynomial division to factor the cubic polynomial.

**Synthetic Division**:
Dividing \( 3x^3 - 14x^2 + 0x + 32 \) by \( x - 2 \):

1. Coefficients of the polynomial: \( [3, -14, 0, 32] \).
2. Perform the synthetic division process:
   - \( 2 \) | \( 3 \)   \( -14 \)   \( 0 \)   \( 32 \)
   - First, bring down the \( 3 \).
   - Multiply \( 2 \times 3 = 6 \), then add to the next coefficient \( -14 + 6 = -8 \).
   - Multiply \( 2 \times -8 = -16 \), then add to the next coefficient \( 0 + (-16) = -16 \).
   - Multiply \( 2 \times -16 = -32 \), then add to the next coefficient \( 32 + (-32) = 0 \).

Result of the division is \( 3x^2 - 8x - 16 \).

Now, factor \( 3x^2 - 8x - 16 \) using the quadratic formula or factoring by grouping.

Quadratic formula:
\[
x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(3)(-16)}}{2(3)} = \frac{8 \pm \sqrt{64 + 192}}{6} = \frac{8 \pm \sqrt{256}}{6} = \frac{8 \pm 16}{6}.
\]

Thus:
\[
x = \frac{8 + 16}{6} = 4 \quad \text{or} \quad x = \frac{8 - 16}{6} = -\frac{4}{3}.
\]

Therefore, \( 3x^2 - 8x - 16 = 3(x - 4)(x + \frac{4}{3}) \).

Thus, the complete factorization is:
\[
3x^3 - 14x^2 + 32 = (x - 2)(3x + 4)(x - 4).
\]

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Would you like to go over this explanation in more detail or continue to the next question? Here are some related questions:

1. Can synthetic division be used for all polynomials?
2. What is the significance of the Factor Theorem?
3. How does the quadratic formula help in solving second-degree polynomials?
4. How can we verify the factorization of a cubic polynomial?
5. Why is \( x - 2 \) chosen as a factor in this problem?

**Tip**: When solving cubic equations, checking for rational roots (like \( x - 2 \)) first can simplify factorization immensely.

1a) Show that x - 2 is a factor of 3x^3 - 14x^2 + 32. 1b) Hence factorise 3x^3 - 14x^2 + 32 completely.

Learn how to determine if x - 2 is a factor of 3x^3 - 14x^2 + 32 using the Factor Theorem and synthetic division. Step-by-step explanation provided, including full factorization of the cubic polynomial.

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