Simplifying Rational Expressions
Because rational expressions are just fractions with variables, the process of simplifying rational expressions is conceptually the same as the process of dividing numeric fractions. You can divide the top and the bottom of the rational expression by the same number. Because dividing the numerator and denominator by the same number is the same as dividing by 1, it leaves you with an equivalent fraction.
The complicated part of simplifying rational expressions is that you often have to factor in order to see what numbers or variables you can divide the numerator and the denominator by. To find factors you can divide by, you can:
- Factor out a constant
- Factor polynomials into binomials
- Use special products to factor trinomials into binomials.
You may do several or all of the above, even multiple times, to fully simplify a rational expression.
Example of factoring out a constant and simplifying:
$$\require{cancel}\eqalign{& \dfrac{7x^2-x}{x^3}\qquad &&\text{ }\\ \\&\dfrac{\overset{x(7x-1)}{\bcancel{7x^2-x}}}{(x)(x)(x)}&&\text{Factor x out of numerator and denominator}\\\\ &\dfrac{\overset{\bcancel{x}(7x-1)}{\bcancel{7x^2-x}}}{\bcancel{(x)}x^2}\qquad &&\text{Cancel x in numerator and denominator}\\ \\&\dfrac{7x-1}{x^2}}$$
Example of factoring into binomials and simplifying:
$$\eqalign{& \dfrac{x^2+x-6}{x-2}\qquad &&\text{ }\\ \\&\dfrac{\overset{(x-2)(x+3)}{\bcancel{x^2+x-6}}}{x-2}&&\text{Factor the numerator into binomials}\\ \\&\dfrac{\overset{\bcancel{(x-2)}(x+3)}{\bcancel{x^2+x-6}}}{\bcancel{x-2}}\qquad &&\text{Cancel out common terms}\\ \\ &\dfrac{x+3}{1}=x+3}$$
Simplifying fractions is just a matter of factoring out common terms among numerators and denominators and then simplifying, just as you would do with regular fractions.