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Definitions and Simplifying
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# Simplifying

Ultimately, reational expressions are nothing more than big fractions. Simplifying rational expressions works exactly the same way as aimplifying (or reducing) fractions. First you factor the numerator and denominator then you cancel any factors that they have in common. The examples show how it works.

# Example 1

Simplify the expression $\frac{x^2 - 2x - 3}{x^2 + x - 12}$

If we factor the numerator and denominator the expression becomes:

$$\frac{(x + 1)(x - 3)}{(x - 3)(x + 4)}$$

Now we can cancel the two x - 3 factors to get the simplified version.

$$\frac{x + 1}{x + 4}$$

# Example 3

Simplify the expression $$\frac{2x - 1}{2x^2 + 3x - 2}$$.

If we factor the numerator and denominator the expression becomes:

$$\frac{2x - 1}{(2x - 1)(x + 2)}$$

Now we can cancel the two 2x - 1 terms but we need to be clear on what's going to be left behind in the numerator since the 2x - 1 term is the only one there. We can't have nothing in the numerator so what's left behind? The answer is a 1. To make that clearer you can think of the expression this way:

$$\frac{1 (2x - 1)}{(2x - 1)(x + 2)}$$

We don't usually write out a 1 like that but, in this case, it helps to make it clearer why that's what's left. Now when we cancel the two 2x - 1 terms, we're left with

$$\frac{1}{x + 2}$$

# Example 2

Simplify the expression $$\frac{x^2 - 6x + 8", "x3 + 2x^2 - 4x - 8"}$$?

If we factor the numerator and denominator the expression becomes:

$$\frac{(x - 2)(x - 4)}{(x - 2)(x + 2)^2}$$

If you're not sure how I factored the denominator the procedure is called factoring by grouping. You can find an explanation of the process in our polynomial short course.

Now we can cancel the two x - 2 terms to get the simplified version.

$$\frac{"x - 4}{(x + 2)^2}=\frac{x - 4}{x^2 + 4x + 4}$$

Which of those versions, the factored or the multiplied out, is the simplified one is a matter of opinion. Some books and instructors prefer the first and others the second.

# Example 4

Simplify the expression $\frac{x^2 - 6x + 8}{x^3+2x^2-4x-8}$.

If we factor the numerator and denominator the expression becomes:

$$\frac{(x - 2)(x - 4)}{(x - 2)(x + 2)^2}$$

If you're not sure how I factored the denominator the procedure is called factoring by grouping. You can find an explanation of the process in our polynomial short course.

Now we can cancel the two x - 2 terms to get the simplified version.

$$\frac{x-4}{(x+2)^2}=\frac{x-4}{x^2+4x+4}$$

Which of those versions, the factored or the multiplied out, is the simplified one is a matter of opinion. Some books and instructors prefer the first and others the second.

Dyanmic Tutorial - Simplifying Rational Expressions

Directions: This solution has 3 steps. To see a description of each step click on the boxes on the left side below. To see the calculations, click on the corresponding box on the right side. Try working out the solution yourself and use the descriptions if you need a hint and the calculations to check your solution.