Adding Fractions

Adding Fractions Numerator and Denominator The number or algebraic expression that appears on the top line of a fraction is called the numerator of the fraction. The number of algebraic expression that appears on the bottom line of a fraction is called the denominator of the fraction.

Adding Fractions Expressed in symbols, the rule for adding fraction is as follows:

Let’s break this down to see everything that is expressed in this rule. The numerator of the sum is a · d + b · c. You can remember the numerator without having to memorize this particular formula by remembering the pattern of cross-multiplying. To create the numerator, you multiply each numerator by the opposing denominator, forming a “cross” pattern. To get the denominator of the sum, you just multiply the two denominators ( b and d ) together. Example Work out each of the following sums of fractions.

Solution

Often it will be possible for you to simplify your fractional expressions by combining “like terms” just as you do when FOILing a polynomial. Although this kind of simplification is not always needed just to get the right answer, if can make your fractional expressions much easier to deal with. Remember to keep the numerator and denominator separate when combining like terms!

In Example (b), note how when the cross-multiplication is done, the “7” from the numerator of the first fraction multiplies the entire quantity ( x + 1) that is in the denominator of the second fraction, not just the x . Also notice that when the two denominators are multiplied to create the denominator of the sum, the “10” from the

denominator of the first fraction multiplies everything (i.e. the entire quantity ( x + 1)) that appears in the denominator of the second fraction.

When simplifying fractions, simplify the numerator and denominator separately. You cannot combine like terms from the numerator with like terms from the denominator (or vice versa). Often you will need to FOIL when simplifying the numerator and denominator of fractions that involve algebraic expressions such as x .

This answer is not the simplest one that is possible. If you look closely at the middle fraction above, you can see that every single term in the numerator has at least one factor of ( x + 1). The denominator also has a factor of ( x + 1). These “common” factors can be factored out of the numerator and the denominator as shown below.

When you have a common factor that you have pulled out of every term in the numerator, and it matches a factor that shows up in the denominator, you can almost always cancel this factor from both the numerator and the denominator.

, provided x

-1.

The only situation when it is not okay to cancel the factor of ( x + 1) from the top and bottom is when you have the x -value of x = - 1 (i.e. the particular x -value that makes the factor of ( x + 1) equal to zero). Adding and Subtracting Rational Expressions With Like Denominators After studying this lesson, you will be able to: Add and subtract rational expressions with like denominators. •

Remember that when we add or subtract fractions, we must have a common denominator before we can add. The same is true for adding and subtracting rational expressions.

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Once we have a common denominator, we keep the denominator and we add the numerators. Collect like terms in the numerator.

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Next, simplify if possible by factoring, canceling, and/or reducing.

Example 1

We have a common denominator so we can add the numerators. This will not reduce so this is the answer.

Example 2

This is a subtraction problem, so we have to "add the opposite" before we do anything else. This means we will take the opposite of everything in the numerator that follows the subtraction sign. Now we are adding and we have a common denominator so we keep the denominator and add the numerators. This will not reduce so this is the answer Example 3 This is a subtraction problem, so we have to "add the opposite" before we do anything else. This means we will take the opposite of everything in the numerator that follows the subtraction sign. Now we are adding and we have a common denominator so we keep the denominator and add the numerators. This will reduce - the numerator will factor. Cancel out the binomials 2 This is the final answer Example 4 This is a subtraction problem, so we have to "add the opposite" before we do anything else. This means we will take the opposite of everything in the numerator that follows the subtraction sign. Now we are adding and we need a common denominator. Factor out -1 from the 2 - x Move the -1 to the top and we will have a common denominator. Distribute the -1 Add the numerators