In this lesson, students will learn how to simplify rational expressions in the numerator and denominator of a fraction by factoring and cancelling like terms.
A rational expression is a fraction whose numerator and denominator are polynomials. They are sometimes referred to as algebraic fractions. Reducing rational expressions is similar to reducing numerical fractions. Factors can be canceled only if they appear as factors of both the numerator and the denominator; they are canceled because they reduce to 1.
However, terms separated by addition or subtraction in the numerator or denominator cannot be canceled.
In review, you can simplify fractions by canceling common factors in the numerator and denominator.
Finding the greatest common factor of a polynomial is a helpful strategy when simplifying algebraic fractions in which common factors appear in both the numerator and denominator.
Suppose you want to factor the following expression:
You can start by writing each term as a product of factors:
You can see that both terms have two 2's and one x in common. Multiplying these common factors together equals 4x.
You can begin to rewrite your expression by writing your greatest common factor, 4x, on the outside of the parentheses, and writing the remaining factors of each term inside the parentheses. From the first term, you have two x's remaining, or x^2, and in your second term, you have a negative and a 2, in other words, -2. This results in the factored form of your expression:
When simplifying rational expressions, you want to start by writing both the numerator and denominator as a products of their prime factors and variable factors.
Suppose you want to simplify the following expression. Begin by rewriting both numerator and denominator as products of their prime factors and variable factors:
You can see that both the numerator and denominator have two 2's and four a's as common factors. You can cancel all of these factors out.
This leaves you with one 3 and two a's multiplied together in the numerator, and one 2 in the denominator. Multiplying your remaining factors provides:
Source: This work is adapted from Sophia author Colleen Atakpu.