Find the factors of `(X^(2)-x-132)`
(A) `(x-11)(x-12)`
(B) `(x+12)(x-11)`
(C ) `(x+11)(x+12)`
(D) `(x-12)(x+11)`

Options

A. D
B. B
C. A
D. C

Answer

To find the factors of the expression \( x^2 - x - 132 \), we can follow these steps:

### Step 1: Identify the quadratic expression
We start with the quadratic expression:
\[ x^2 - x - 132 \]

### Step 2: Multiply the coefficient of \( x^2 \) by the constant term
The coefficient of \( x^2 \) is 1, and the constant term is -132. We multiply these:
\[ 1 \times -132 = -132 \]

### Step 3: Find two numbers that multiply to -132 and add to -1
We need to find two numbers that multiply to -132 and add to the coefficient of \( x \), which is -1. The numbers that satisfy these conditions are 11 and -12:
\[ 11 \times -12 = -132 \quad \text{and} \quad 11 + (-12) = -1 \]

### Step 4: Rewrite the middle term using the two numbers
We can rewrite the expression \( x^2 - x - 132 \) by splitting the middle term:
\[ x^2 + 11x - 12x - 132 \]

### Step 5: Group the terms
Now we group the terms:
\[ (x^2 + 11x) + (-12x - 132) \]

### Step 6: Factor by grouping
Now we factor out the common factors from each group:
\[ x(x + 11) - 12(x + 11) \]

### Step 7: Factor out the common binomial
Now we can factor out the common binomial \( (x + 11) \):
\[ (x + 11)(x - 12) \]

### Conclusion
Thus, the factors of the expression \( x^2 - x - 132 \) are:
\[ (x + 11)(x - 12) \]

### Final Answer
The correct option is (D) \( (x - 12)(x + 11) \).