Find the common zeroes of the polynomials`x^(3)+x^(2)-2x-2` and `x^(3)-x^(2)-2x+2`.

Explanation

To find the common zeroes of the polynomials \( P(x) = x^3 + x^2 - 2x - 2 \) and \( Q(x) = x^3 - x^2 - 2x + 2 \), we will follow these steps:

### Step 1: Find the SCF (Highest Common Factor)
We will use the long division method to find the SCF of the two polynomials.

1. **Divide \( P(x) \) by \( Q(x) \)**:
- The leading term of both polynomials is \( x^3 \). When we divide \( P(x) \) by \( Q(x) \), we start with \( 1 \) as the first term of the quotient.
- Multiply \( Q(x) \) by \( 1 \):
\[ Q(x) \cdot 1 = x^3 - x^2 - 2x + 2 \]
- Subtract this from \( P(x) \):
\[ P(x) - (x^3 - x^2 - 2x + 2) = (x^2 + x^2) + (2x + 2x) - 2 - 2 = 2x^2 + 4 \]

### Step 2: Simplify the Result
Now we have:
\[ R(x) = 2x^2 + 4 \]
Next, we will find the SCF of \( Q(x) \) and \( R(x) \).

### Step 3: Divide \( Q(x) \) by \( R(x) \)
1. **Divide \( Q(x) \) by \( R(x) \)**:
- The leading term of \( Q(x) \) is \( x^3 \) and the leading term of \( R(x) \) is \( 2x^2 \). The first term of the quotient is \( \frac{1}{2}x \).
- Multiply \( R(x) \) by \( \frac{1}{2}x \):
\[ R(x) \cdot \frac{1}{2}x = x(2x^2 + 4) = x^3 + 2x \]
- Subtract this from \( Q(x) \):
\[ Q(x) - (x^3 + 2x) = (-x^2 - 2x + 2) - (2x) = -x^2 - 4 \]

### Step 4: Find the SCF
Now we have:
\[ S(x) = -x^2 - 4 \]
Next, we will find the SCF of \( R(x) \) and \( S(x) \).

### Step 5: Divide \( R(x) \) by \( S(x) \)
1. **Divide \( R(x) \) by \( S(x) \)**:
- The leading term of \( R(x) \) is \( 2x^2 \) and the leading term of \( S(x) \) is \( -x^2 \). The first term of the quotient is \( -2 \).
- Multiply \( S(x) \) by \( -2 \):
\[ S(x) \cdot (-2) = 2x^2 + 8 \]
- Subtract this from \( R(x) \):
\[ R(x) - (2x^2 + 8) = (2x^2 + 4) - (2x^2 + 8) = -4 \]

### Step 6: Determine the Common Zeroes
The SCF of the two polynomials is \( -4 \), which means the common factor is \( 2(x^2 + 2) \).

### Step 7: Find the Zeroes of the Common Factor
To find the zeroes of \( x^2 + 2 \):
\[ x^2 + 2 = 0 \implies x^2 = -2 \implies x = \pm \sqrt{-2} = \pm i\sqrt{2} \]

### Final Answer
The common zeroes of the polynomials \( P(x) \) and \( Q(x) \) are:
\[ x = i\sqrt{2} \quad \text{and} \quad x = -i\sqrt{2} \]