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Canceling Common Terms in Algebraic Fractions

Canceling Common Terms in Algebraic Fractions

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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.

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Tutorial
This tutorial covers how to cancel common terms in algebraic fractions, through the definition and discussion of:
  1. Rational Expressions
  2. Simplifying Fractions
  3. Greatest Common Factor in Polynomials
  4. 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.
fraction numerator left parenthesis x plus 3 right parenthesis over denominator left parenthesis x plus 3 right parenthesis end fraction equals 1

However, terms separated by addition or subtraction in the numerator or denominator cannot be canceled.

Consider the expression below.
fraction numerator x minus 2 over denominator x plus 5 end fraction not equal to fraction numerator negative 2 over denominator 5 end fraction

You can verify this because if you substituted a value of 3 in for both xs, you’d have the following, which is not -2 over 5.
fraction numerator 3 minus 2 over denominator 3 plus 5 end fraction equals 1 over 8 not equal to fraction numerator negative 2 over denominator 5 end fraction


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: Write the numerator and denominator as products of prime factors:

File:1259-cancel1.PNG

Step 2: 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.

File:1260-cancel2.PNG

Step 3: Multiply remaining factors in the numerator and denominator. In the numerator you have three 2s remaining, and in the denominator, you have one 3 and one 5, so multiply accordingly and simplify.

File:1261-cancel3.PNG

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 expression:

4 x cubed minus 8 x

You can start by writing each term as a product of factors.

4 x cubed minus 8 x equals 2 times 2 times x times x times x minus 2 times 2 times 2 times x

You can see that both terms have two 2s and one x in common. Multiplying these common factors together equals 4x.

File:1262-cancel4.PNG

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 xs 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 x left parenthesis x squared minus 2 right parenthesis

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:

fraction numerator 12 a to the power of 6 over denominator 8 a to the power of 4 end fraction equals fraction numerator 2 times 2 times 3 times a times a times a times a times a times a over denominator 2 times 2 times 2 times a times a times a times a end fraction

You can see that both the numerator and denominator have two 2s and four as as common factors. You can cancel all of these factors out.

File:1263-cancel5.PNG

This leaves you with one 3 and two as multiplied together in the numerator, and one 2 in the denominator. Multiplying your remaining factors provides:

fraction numerator 3 times a times a over denominator 2 end fraction equals fraction numerator 3 a squared over denominator 2 end fraction
Simplify the following expression, beginning by factoring both the numerator and denominator:
fraction numerator 7 x plus 21 over denominator 2 x plus 6 end fraction equals fraction numerator 7 times x plus 7 times 3 over denominator 2 times x plus 2 times 3 end fraction
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.

File:1264-cancel6.PNG

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:

File:1265-cancel7.PNG

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 find 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.

Source: This work is adapted from Sophia author Colleen Atakpu.