Given that `x-sqrt(5)` is a factor of the cubic polynomial `x^(3)-3sqrt(5)x^(2)+13x-3sqrt(5)` , find all the zeroes of the polynomial.

Answer

To find all the zeroes of the polynomial \( P(x) = x^3 - 3\sqrt{5}x^2 + 13x - 3\sqrt{5} \), given that \( x - \sqrt{5} \) is a factor, we will perform polynomial long division and then solve the resulting quadratic equation.

### Step 1: Polynomial Long Division
We will divide \( P(x) \) by \( x - \sqrt{5} \).

1. **Divide the leading term**:
\[ \frac{x^3}{x} = x^2 \]

2. **Multiply** \( x^2 \) by \( x - \sqrt{5} \):
\[ x^2(x - \sqrt{5}) = x^3 - \sqrt{5}x^2 \]

3. **Subtract** this from \( P(x) \):
\[ P(x) - (x^3 - \sqrt{5}x^2) = (-3\sqrt{5} + \sqrt{5})x^2 + 13x - 3\sqrt{5} = -2\sqrt{5}x^2 + 13x - 3\sqrt{5} \]

4. **Next term**: Divide the leading term:
\[ \frac{-2\sqrt{5}x^2}{x} = -2\sqrt{5}x \]

5. **Multiply**:
\[ -2\sqrt{5}x(x - \sqrt{5}) = -2\sqrt{5}x^2 + 10 \]

6. **Subtract**:
\[ (-2\sqrt{5}x^2 + 13x - 3\sqrt{5}) - (-2\sqrt{5}x^2 + 10) = 13x - 3\sqrt{5} - 10 = 13x - 10 - 3\sqrt{5} \]

7. **Next term**: Divide the leading term:
\[ \frac{(13x - 10 - 3\sqrt{5})}{x} = 3 \]

8. **Multiply**:
\[ 3(x - \sqrt{5}) = 3x - 3\sqrt{5} \]

9. **Subtract**:
\[ (13x - 10 - 3\sqrt{5}) - (3x - 3\sqrt{5}) = 10x - 10 = 0 \]

Since the remainder is zero, we have:
\[ P(x) = (x - \sqrt{5})(x^2 - 2\sqrt{5}x + 3) \]

### Step 2: Solve the Quadratic Equation
Now we need to find the roots of the quadratic \( x^2 - 2\sqrt{5}x + 3 = 0 \).

Using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1 \), \( b = -2\sqrt{5} \), and \( c = 3 \).

1. **Calculate the discriminant**:
\[ b^2 - 4ac = (-2\sqrt{5})^2 - 4 \cdot 1 \cdot 3 = 20 - 12 = 8 \]

2. **Substitute into the formula**:
\[ x = \frac{2\sqrt{5} \pm \sqrt{8}}{2} \]
\[ = \frac{2\sqrt{5} \pm 2\sqrt{2}}{2} \]
\[ = \sqrt{5} \pm \sqrt{2} \]

### Step 3: Final Roots
Thus, the zeroes of the polynomial are:
1. \( x = \sqrt{5} \)
2. \( x = \sqrt{5} + \sqrt{2} \)
3. \( x = \sqrt{5} - \sqrt{2} \)