This tutorial will cover how to cancel common terms in algebraic fractions, through the definition and discussion of:
- Rational Expressions
- Simplifying Fractions
- Greatest Common Factor in Polynomials
- Simplifying Rational Expressions
1. Rational Expressions
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.
You can see how the fraction below simplifies to 1.
However, terms separated by addition or subtraction in the numerator or denominator cannot be canceled.
Consider the expression below.
- You can verify this because if you substituted a value 3 in for both x's, you'd have the following, which is not -2 over 5.
2. Simplifying Fractions
In review, you can simplify fractions by canceling common factors in the numerator and denominator.
Suppose you want to simplify the following fraction.
Step 1: The first step is to write the numerator and denominator as products of prime factors:
Step 2: The second step is to cancel factors that appear in both the numerator and denominator. You can cancel out one 2 and one 3 in both the numerator and denominator:
Step 3: The third step is to multiply remaining factors in the numerator and denominator. In the numerator you have three 2's remaining, and in the denominator, you have one 3 and one 5, so multiply accordingly and simplify:
Simplifying rational expressions is similar to reducing numerical fractions, because you identify common factors to cancel.
3. Greatest Common Factor in Polynomials
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:
4. Simplifying Rational Expressions
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:
Simplify the following expression, beginning by factoring both the numerator and denominator:
- Looking at your numerator, you can see that you have a common factor of 7. Therefore, you can factor out a 7 and rewrite your expression with the remaining factors, x and plus 3. Next, looking at your denominator, you can see that you have a common factor of 2 in both terms, so, again, you can factor out the 2 by writing it on the outside of the parentheses and writing your remaining factors, x and plus 3, inside the parentheses.
- Now you can see that you have a common factor of x plus 3 in both the numerator and the denominator, which can be canceled out, leaving you with your final expression:
Today you learned the definition of a rational expression, or algebraic fraction, which is a fraction whose numerator and denominator are polynomials. You also reviewed how to simplify fractions and finding the greatest common factor in polynomials. Lastly, you learned how to simplify rational expressions, by 1) writing the numerator and denominator as products of prime factors, 2) canceling factors that appear in both the numerator and denominator, and 3) multiplying the remaining factors in the numerator and denominator.