(a) Use factorization to simplify the given expression.

*x*2 + *x* − 6*/x*2 − 5*x* + 6

To simplify the expression \(\frac{x^2 + x - 6}{x^2 - 5x + 6}\) using factorization, follow these steps:

### 1. Factorize the Numerator
The numerator is \(x^2 + x - 6\). We need to find two numbers that multiply to \(-6\) (the constant term) and add to \(1\) (the coefficient of \(x\)).

The numbers \(3\) and \(-2\) work because:
\[
3 \times -2 = -6
\]
\[
3 + (-2) = 1
\]

So, the numerator can be factorized as:
\[
x^2 + x - 6 = (x + 3)(x - 2)
\]

### 2. Factorize the Denominator
The denominator is \(x^2 - 5x + 6\). We need to find two numbers that multiply to \(6\) (the constant term) and add to \(-5\) (the coefficient of \(x\)).

The numbers \(-3\) and \(-2\) work because:
\[
-3 \times -2 = 6
\]
\[
-3 + (-2) = -5
\]

So, the denominator can be factorized as:
\[
x^2 - 5x + 6 = (x - 3)(x - 2)
\]

### 3. Simplify the Expression
Now, the expression becomes:
\[
\frac{(x + 3)(x - 2)}{(x - 3)(x - 2)}
\]

Notice that \((x - 2)\) appears in both the numerator and the denominator, so it can be canceled out (as long as \(x \neq 2\), because division by zero is undefined).

Thus, the simplified expression is:
\[
\frac{x + 3}{x - 3}, \quad \text{where } x \neq 2 \text{ and } x \neq 3.
\]

Would you like further details or have any questions?

Here are 5 related questions:
1. How do you factor trinomials of the form \(ax^2 + bx + c\)?
2. What happens if the numerator and denominator don't share any common factors?
3. How do you simplify expressions with higher degree polynomials?
4. What are the conditions under which terms can be canceled in a rational expression?
5. How do restrictions on the variable \(x\) affect the final simplified expression?

**Tip:** Always check for common factors in both the numerator and the denominator to simplify the expression efficiently.

Learn how to simplify the fraction (x^2 + x - 6) / (x^2 - 5x + 6) using factorization. This problem covers algebraic concepts, including quadratic equations and fraction simplification. Suitable for students in Grades 7-9.

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