The expression `((x-1)(x-2)(x^(2) - 9x + 14))/((x-7)(x^(2) - 3x + 2))` in the lowest terms is:

Options

A. `x+2`
B. `x-3`
C. `x+3`
D. `x-2`

Answer

To simplify the expression \(\frac{(x-1)(x-2)(x^2 - 9x + 14)}{(x-7)(x^2 - 3x + 2)}\) 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 the denominator.

1. **Numerator: \(x^2 - 9x + 14\)**
- 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 since \(-7 \times -2 = 14\) and \(-7 + (-2) = -9\).
- Thus, we can factor it as:
\[ x^2 - 9x + 14 = (x - 7)(x - 2) \]

2. **Denominator: \(x^2 - 3x + 2\)**
- We look for two numbers that multiply to \(2\) and add up to \(-3\).
- The numbers \(-1\) and \(-2\) work since \(-1 \times -2 = 2\) and \(-1 + (-2) = -3\).
- Thus, we can factor it as:
\[ x^2 - 3x + 2 = (x - 1)(x - 2) \]

### Step 2: Rewrite the Expression
Now we can rewrite the original expression using the factored forms:
\[ \frac{(x-1)(x-2)(x-7)(x-2)}{(x-7)(x-1)(x-2)} \]

### Step 3: Cancel Common Factors
Next, we can cancel the common factors in the numerator and denominator:
- The factors \((x - 7)\), \((x - 1)\), and \((x - 2)\) appear in both the numerator and the denominator.

After canceling, we are left with:
\[ \frac{(x - 2)}{1} = x - 2 \]

### Final Answer
Thus, the expression in its lowest terms is:
\[ \boxed{x - 2} \]

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