(x^2+x-6)/(x-2)

2 min read Jun 17, 2024

Simplifying the Rational Expression (x^2 + x - 6)/(x - 2)

This article explores the simplification of the rational expression (x^2 + x - 6)/(x - 2). We will break down the process step-by-step to understand how to simplify this expression and identify any restrictions on the variable.

Step 1: Factor the Numerator

The numerator, x^2 + x - 6, can be factored into two binomials.

We are looking for two numbers that multiply to -6 and add up to 1 (the coefficient of the x term).
The numbers 3 and -2 satisfy these conditions: 3 * (-2) = -6 and 3 + (-2) = 1.

Therefore, we can factor the numerator as (x + 3)(x - 2).

Step 2: Simplify the Expression

Now we have the expression: [(x + 3)(x - 2)] / (x - 2).

Since the factor (x - 2) appears in both the numerator and denominator, we can cancel them out, resulting in:

(x + 3)

Step 3: Identify Restrictions

While the simplified expression is (x + 3), we need to remember the original expression. The original expression has a denominator of (x - 2). This denominator cannot be equal to zero because division by zero is undefined.

Therefore, the restriction on the variable x is: x ≠ 2

Conclusion

The simplified form of the rational expression (x^2 + x - 6)/(x - 2) is (x + 3), with the restriction that x ≠ 2.