When simplifying rational expressions, it is crucial to distinguish between factors and terms. You may only remove common factors, which are parts of a product, from the numerator and denominator. You cannot remove a term from a sum. For example, in the rational expression $$\frac{x+5}{x}$$, the $$x$$ in the numerator is a term in a sum, not a factor. Therefore, it cannot be canceled with the $$x$$ in the denominator. Attempting to do so is a mathematical error, equivalent to incorrectly canceling the $$2$$'s in the numerical fraction $$\frac{2+5}{2}$$.

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To simplify a rational expression, follow these steps:
1. Factor the numerator and denominator completely.
2. Simplify by dividing out common factors using the Equivalent Fractions Property.

Note that factors are multiplied to make a product, so you can remove a common factor from a product. However, you cannot remove a term from a sum.

Procedure for Simplifying a Rational Expression

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OpenStax Intermediate Algebra

To simplify the rational expression $$\frac{x^2+5x+6}{x^2+8x+12}$$, follow the standard procedure:
Step 1: Factor the numerator and denominator completely. The numerator $$x^2+5x+6$$ factors into $$(x+2)(x+3)$$. The denominator $$x^2+8x+12$$ factors into $$(x+2)(x+6)$$.
Step 2: Simplify by dividing out the common factor $$(x+2)$$. The result is $$\frac{x+3}{x+6}$$.

Example of Simplifying a Rational Expression

Canceling Common Factors versus Terms in Rational Expressions

To simplify a rational expression where the terms have a greatest common factor (GCF), first factor out the GCF from both the numerator and the denominator, and then factor the remaining polynomials completely. Finally, divide out the common factors.

For instance, consider the expression:
$$\frac{3a^2 - 12ab + 12b^2}{6a^2 - 24b^2}$$

First, factor out the GCF from the numerator (which is $$3$$) and the denominator (which is $$6$$):
$$\frac{3(a^2 - 4ab + 4b^2)}{6(a^2 - 4b^2)}$$

Next, factor the resulting perfect square trinomial in the numerator and the difference of squares in the denominator:
$$\frac{3(a - 2b)(a - 2b)}{6(a + 2b)(a - 2b)}$$

Rewrite $$6$$ as $$3 \cdot 2$$ to clearly identify all numerical common factors:
$$\frac{3(a - 2b)(a - 2b)}{3 \cdot 2(a + 2b)(a - 2b)}$$

Remove the common factors of $$3$$ and $$a - 2b$$ to obtain the simplified expression:
$$\frac{a - 2b}{2(a + 2b)}$$

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