Simplifying Rational Expressions
April 23, 2025
Just like in algebraic expressions, we can simplify rational expressions by factoring. Since these expressions are written as fractions (with a numerator and a denominator), we look for common factors in both parts. After factoring, we cancel out any common factors—as long as they don’t make the denominator zero.
A number in the fraction form \dfrac{X}{Y}, with Y \neq 0, is said to be in simplified form (or lowest terms), if the greatest common factor of the numerator and denominator is 1.
Similar with regular fractions, rational expressions can be simplified by canceling common factors in the numerator and the denominator.
- Not yet simplified: \dfrac{6}{9}, both numerator and denominator have a common factor of 3.
- Simplified form: \dfrac{6 ÷ 3}{9 ÷ 3} = \dfrac{2}{3}
We’ll use the same idea when simplifying rational expressions using the property below.
Suppose a rational expression is in the form \dfrac{P}{Q}, where both P \text{ and } Q are polynomials and Y \neq 0. If K is any nonzero polynomial, multiplying both the numerator and the denominator of a rational expression by K does not change its value:
Before we proceed further, let us be reminded on how to factor expressions completely, using this guidelines:
To factor expressions completely, here are some steps you must follow:
- Start with factoring by greatest common monomial factor, if possible.
- If both the exponents are even, the constants are perfect squares, and in the form of subtraction, it is factorable by difference of two squares.
- Similarly, if the exponents are multiples of 3, the constants are perfect cubes, then it is factorable by sum/difference of two cubes.
- If there are three terms present, evaluate if it is a perfect square trinomial.
- If there are three terms present, and the expression is not a perfect square trinomial, consider factoring by grouping
Now that we understand the property, we can utilize these steps to simplify rational expressions:
Step 1: Factor out the numerator and denominator completely, and;
Step 2: Cancel out all the common factors present in the rational expression
Simplify \bf\dfrac{3x^2 - 48}{5x + 20}
Therefore, \boxed{\dfrac{3x^2 - 48}{5x + 20} = \dfrac{3(x - 4)}{5}}.
Simplify \bf \dfrac{x^2 - 8}{x^2 - 4}
Therefore, \boxed{\dfrac{x^3 - 8}{x^2 - 4} = \dfrac{x^2 + 2x + 4}{x + 2}}.
Simplify \bf \dfrac{x^3 + 2x^2 -x - 2}{x^2 + 2x - 3}
Therefore, \boxed{\dfrac{x^3 + 2x^2 - x - 2}{x^2 + 2x - 3} = \dfrac{(x + 2)(x + 1)}{x + 3}}.
Simplify \bf \dfrac{2x^2 + 5x + 2x + 5}{x^2 + 3x + 2}
Therefore, \boxed{\frac{2x^2 + 5x + 2x + 5}{x^2 + 3x + 2} = \frac{2x + 5}{x + 2}}.
Simplify \bf \dfrac{x^2 - 9}{x^2 + 6x + 9}
Therefore, \boxed{\dfrac{x^2 - 9}{x^2 + 6x +9} = \dfrac{x - 3}{x + 3}}.
