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Adding and Subtracting Rational Expressions

Adding and Subtracting Rational Expressions

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Tutorial


  • Adding and Subtracting Numeric Fractions
  • Adding and Subtracting Rational Expressions with a Common Denominator
  • Adding and Subtracting Rational Expressions with an Uncommon Denominator

Adding and Subtracting Numeric Fractions

Reviewing how to add and subtract numeric fractions is helpful when learning how to add and subtract algebraic fractions.  This is because the same general principle is applied in both cases.  In order to add two fractions, we want to express each fraction, such that the denominators are the same.  Then we can simply add or subtract across the numerators, and retain the common denominator. 

When adding 5 over 7 plus 3 over 7 this is fairly straightforward.  The denominators are already equivalent, so we can just add 5 and 3 to get a sum of 8 over 7

When the denominators are not the same, we need to write equivalent equations with the same denominator.  

Consider the subtraction problem: 5 over 6 minus 1 half

Adding Rational Expression with Common Denominators

When the denominators between two factions are the same, adding and subtracting is much easier.  This is also the case with rational expressions.  Below is an example of adding rational expressions with the same denominator:

 


Adding and subtracting rational expressions with common denominators operates in the same way as adding and subtracting numeric fractions with common denominators.  We add or subtract across the numerators, and keep the denominator the same. 


Adding and Subtracting Rational Expressions with Uncommon Denominators

Here comes the most complicated part about the entire lesson.  When adding or subtracting rational expressions without a common denominator, we must re-express the fractions so that they do have a common denominator.  That way, we can add or subtract as we did in the example above.

To find a common denominator between two algebraic fractions:

  • Write the prime factorizations of each denominator (similar to what we did in our numeric example)
  • Determine the greatest number of times a particular factor appears within all factorizations.
  • Include that quantity (from the previous step) to be multiplied to find the common denominator

The last step is probably the most confusing, so we will be sure to tackle an example of this below:

Let's solve this subtraction problem: fraction numerator x plus 2 over denominator 2 x squared end fraction minus fraction numerator 7 over denominator x squared plus x end fraction

Our first task is to find a common denominator.  One strategy would be to multiply the two denominators, however this can get messy, because there would be common factors in the numerator and denominator that should be canceled, but not necessary easy to spot.  For this reason, we will factorize each denominator:

2 x squared equals 2 • x • x

x squared plus x equals x left parenthesis x plus 1 right parenthesis

Next, we look at each factor in the factorizations, and determine the greatest number of times it appears between both factorizations:

  • The factor 2 appears only once.  We will include one factor of 2 in the common denominator.
  • The factor x appears at most two times in both factorizations.  We will include two factors of x in the common denominator.
  • The factor (x + 1) appears only once.  We will include one factor of (x + 1) in the common denominator.

We multiply these quantities together to get the common denominator: 2 • x • x left parenthesis x plus 1 right parenthesis space equals space 2 x squared left parenthesis x plus 1 right parenthesis equals 2 x cubed plus 2 x squared

Now that we have a common denominator, we must adjust each numerator so that the actually quantities remain equivalent.  To do this, we attach factors to the numerator that have been attached to the denominator when rewriting the original denominator into the common denominator:



If you are having trouble figuring out which factors you need to multiply into the numerator and denominator, it might help to keep the common denominator written out in its prime factorization.  That way, you can more clearly see which factors are already included in the original fraction, and which factors are yet to be included. 


Now we are ready to subtract, working primarily with the numerators, since the denominators are now the same: