The GCD of `[x^(2)-ax-(a+1)] and [ax^(2)-x-(a+1)]` is :

Options

A. `(x-1)`
B. `(x+1)`
C. `(x+a+1)`
D. `(x-a-1)`

Answer

To find the GCD (Greatest Common Divisor) of the two polynomials \( P_1(x) = x^2 - ax - (a + 1) \) and \( P_2(x) = ax^2 - x - (a + 1) \), we will follow these steps:

### Step 1: Factor the first polynomial \( P_1(x) \)

We start with:
\[ P_1(x) = x^2 - ax - (a + 1) \]

To factor this polynomial, we can look for two numbers that multiply to \(- (a + 1)\) and add to \(-a\). We can manipulate the expression as follows:

1. Rewrite \( P_1(x) \):
\[ P_1(x) = x^2 - ax - a - 1 \]

2. We can rearrange it:
\[ P_1(x) = x^2 + (1 - a)x - (a + 1) \]

3. Now we can factor by grouping or using the quadratic formula if necessary, but let's try to find the roots directly:
\[ P_1(x) = (x - (a + 1))(x + 1) \]

### Step 2: Factor the second polynomial \( P_2(x) \)

Next, we factor the second polynomial:
\[ P_2(x) = ax^2 - x - (a + 1) \]

1. Rewrite \( P_2(x) \):
\[ P_2(x) = ax^2 - x - a - 1 \]

2. Rearranging gives:
\[ P_2(x) = ax^2 + (-1)x - (a + 1) \]

3. We can factor this polynomial as well:
\[ P_2(x) = (x - (a + 1))(ax + 1) \]

### Step 3: Identify common factors

Now we have:
\[ P_1(x) = (x - (a + 1))(x + 1) \]
\[ P_2(x) = (x - (a + 1))(ax + 1) \]

The common factor in both factorizations is:
\[ x + 1 \]

### Conclusion

Thus, the GCD of the two polynomials \( P_1(x) \) and \( P_2(x) \) is:
\[ \text{GCD}(P_1, P_2) = x + 1 \]