Given that `sqrt(2)` is a zero of a polynomial `6x^(3)+sqrt(2x^(2))-10x-4sqrt(2)`, find the other two zeroes.

Explanation

To find the other two zeroes of the polynomial \( f(x) = 6x^3 + \sqrt{2}x^2 - 10x - 4\sqrt{2} \) given that \( \sqrt{2} \) is a zero, we can follow these steps:

### Step 1: Write down the polynomial and identify the factor
Since \( \sqrt{2} \) is a zero of the polynomial, we can say that \( x - \sqrt{2} \) is a factor of \( f(x) \).

### Step 2: Perform polynomial long division
We will divide the polynomial \( f(x) \) by \( x - \sqrt{2} \) using polynomial long division.

1. **Divide the leading term**: Divide \( 6x^3 \) by \( x \) to get \( 6x^2 \).
2. **Multiply**: Multiply \( 6x^2 \) by \( (x - \sqrt{2}) \) to get \( 6x^3 - 6\sqrt{2}x^2 \).
3. **Subtract**: Subtract \( (6x^3 - 6\sqrt{2}x^2) \) from \( f(x) \):
\[ (6x^3 + \sqrt{2}x^2 - 10x - 4\sqrt{2}) - (6x^3 - 6\sqrt{2}x^2) = (1 + 6)\sqrt{2}x^2 - 10x - 4\sqrt{2} = 7\sqrt{2}x^2 - 10x - 4\sqrt{2} \]

### Step 3: Continue the division
1. **Divide the new leading term**: Divide \( 7\sqrt{2}x^2 \) by \( x \) to get \( 7\sqrt{2}x \).
2. **Multiply**: Multiply \( 7\sqrt{2}x \) by \( (x - \sqrt{2}) \) to get \( 7\sqrt{2}x^2 - 14 \).
3. **Subtract**: Subtract \( (7\sqrt{2}x^2 - 14) \) from the previous remainder:
\[ (7\sqrt{2}x^2 - 10x - 4\sqrt{2}) - (7\sqrt{2}x^2 - 14) = -10x + 14 - 4\sqrt{2} = -10x + 14 - 4\sqrt{2} \]

### Step 4: Final division step
1. **Divide the new leading term**: Divide \( -10x \) by \( x \) to get \( -10 \).
2. **Multiply**: Multiply \( -10 \) by \( (x - \sqrt{2}) \) to get \( -10x + 10\sqrt{2} \).
3. **Subtract**: Subtract \( (-10x + 10\sqrt{2}) \) from the previous remainder:
\[ (-10x + 14 - 4\sqrt{2}) - (-10x + 10\sqrt{2}) = 14 - 4\sqrt{2} - 10\sqrt{2} = 14 - 14\sqrt{2} \]

### Step 5: Write the quotient
The quotient from the division is \( 6x^2 + 7\sqrt{2}x - 10 \).

### Step 6: Factor the quadratic
Now we need to factor \( 6x^2 + 7\sqrt{2}x - 10 \). We can use the middle-term splitting method or the quadratic formula.

1. **Using the quadratic formula**:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 6, b = 7\sqrt{2}, c = -10 \).

Calculate the discriminant:
\[ b^2 - 4ac = (7\sqrt{2})^2 - 4 \cdot 6 \cdot (-10) = 98 + 240 = 338 \]

Now substitute into the formula:
\[ x = \frac{-7\sqrt{2} \pm \sqrt{338}}{12} \]

### Step 7: Find the other two zeroes
The two zeroes of the polynomial are:
1. \( x = \sqrt{2} \) (given)
2. \( x = \frac{-7\sqrt{2} + \sqrt{338}}{12} \)
3. \( x = \frac{-7\sqrt{2} - \sqrt{338}}{12} \)