The expression `((x+1)(x-2)(x^(2)-9x+14))/((x-7)(x^(2)-4))` in the lowest terms is

Options

A. x + 2
B. x - 3
C. x + 3
D. None of these

Answer

To simplify the expression \(\frac{(x+1)(x-2)(x^{2}-9x+14)}{(x-7)(x^{2}-4)}\) to its lowest terms, we will follow these steps:

### Step 1: Factor the Quadratic Expressions
First, we need to factor the quadratic expressions in the numerator and denominator.

1. **Factor \(x^2 - 9x + 14\)**:
- We look for two numbers that multiply to \(14\) (the constant term) and add up to \(-9\) (the coefficient of \(x\)).
- The numbers \(-7\) and \(-2\) work because \(-7 \cdot -2 = 14\) and \(-7 + -2 = -9\).
- Thus, we can factor \(x^2 - 9x + 14\) as \((x - 7)(x - 2)\).

2. **Factor \(x^2 - 4\)**:
- This is a difference of squares, which can be factored using the identity \(a^2 - b^2 = (a - b)(a + b)\).
- Here, \(x^2 - 4 = (x - 2)(x + 2)\).

### Step 2: Rewrite the Expression
Now, substituting the factored forms back into the expression, we have:

\[ \frac{(x+1)(x-2)(x-7)(x-2)}{(x-7)(x-2)(x+2)} \]

### Step 3: Cancel Common Factors
Next, we can cancel the common factors in the numerator and denominator:

- The factor \((x - 7)\) appears in both the numerator and denominator.
- The factor \((x - 2)\) also appears in both the numerator and denominator.

After canceling, we get:

\[ \frac{(x + 1)(x - 2)}{(x + 2)} \]

### Step 4: Final Expression
Thus, the expression in its lowest terms is:

\[ \frac{(x + 1)}{(x + 2)} \]