%2F(x-3).jpeg "(x^2-x-6)/(x-3)")
Simplifying the Rational Expression (x^2 - x - 6)/(x - 3)
This article will guide you through the process of simplifying the rational expression (x^2 - x - 6)/(x - 3). We will explore the steps involved and understand the limitations of the simplification.
Factoring the Numerator
The first step is to factor the numerator, which is a quadratic expression. We aim to find two numbers that add up to -1 (the coefficient of the x term) and multiply to -6 (the constant term). These numbers are -3 and 2:
(x^2 - x - 6) = (x - 3)(x + 2)
Simplifying the Expression
Now, we can rewrite the original expression with the factored numerator:
(x^2 - x - 6)/(x - 3) = ((x - 3)(x + 2))/(x - 3)
Since (x - 3) appears in both the numerator and denominator, we can cancel them out as long as x ≠ 3. This is because dividing by zero is undefined.
Therefore, the simplified expression is:
(x + 2), x ≠ 3
Understanding the Restriction
It's crucial to note the restriction x ≠ 3. This means that the simplified expression is equivalent to the original expression for all values of x except for x = 3. At x = 3, the original expression is undefined because it results in division by zero.
In conclusion, the simplified form of (x^2 - x - 6)/(x - 3) is (x + 2) for all values of x except for x = 3.