


Enter an expression containing the sum or difference of fractions and click the Add Fractions button. Algebraic FractionsYou have encountered fractions many times since early in the study of mathematics. They occur in formulas and in many daytoday practical problems. However, the fractions of arithmetic are made up strictly of numbers. We will now study operations on fractions whose components are algebraic expressions. SOLVING EQUATIONS INVOLVING SIGNED NUMBERSOBJECTIVESUpon completing this section you should be able to:
An algebraic fraction is the indicated ratio of two algebraic expressions. In your study of arithmetic you were instructed that fractional answers were always to be left in reduced, or simplified form. For the fraction you "reduced" it to by dividing the numerator and the denominator by 4. The fraction cannot be reduced because no number (other than 1) will divide both numerator and denominator. In simplifying fractions this way you were using the following definition. A fraction is in simplified (or reduced) form if the numerator and denominator contain no common factor (other than 1).
To obtain the simplified form of a fraction apply the following rule. To simplify a fraction factor the numerator and denominator completely and then divide both numerator and denominator by all common factors.
Next divide by the common factors, giving
Now divide by the common factor (x + 2) in both numerator and denominator to get
Notice that even though we were able to factor the numerator and denominator, we still cannot divide since no factors are common to both. The given fraction is already in simplified form. The fact that a given fraction might require any of the methods of factoring you have studied emphasizes again the importance of being proficient in factoring. Solution Here you may use "trial and error" for the numerator and "grouping" for the denominator.
Solution This type of problem requires special attention because it is a common cause for error. At first glance the factors might be mistakenly considered as common, or the fraction might be mistakenly considered as already simplified. Note that the factors cannot be divided since the signs keep them from being identical. If, however, negative 1 is factored from one of the factors, then there are like factors and division can be accomplished.
MULTIPLICATION OF ALGEBRAIC FRACTIONSOBJECTIVESUpon completing this section you should be able to:
An algebraic fraction is the indicated ratio of two algebraic expressions. is the definition of the product of two fractions. In words this says, "multiply numerator by numerator and denominator by denominator." You have used this rule many times in arithmetic as you multiplied fractions. However, remember that all fractional answers must be in simplified form. We could follow the above definition and then simplify the answer as in the previous section. But with algebraic fractions this can lead to very difficult expressions. The following rule allows us to simplify as we multiply, so the answer will then be in simplified form. When multiplying algebraic fractions, factor all numerators and denominators completely, then divide by all factors common to a numerator and denominator before multiplying. The product of the remaining factors of the numerator will be the numerator of the answer and the product of the remaining factors of the denominator will be the denominator of the answer.
DIVISION OF ALGEBRAIC FRACTIONSOBJECTIVESUpon completing this section you should be able to:
Division of fractions is defined in terms of multiplication. To divide multiply by the inverse of the divisor. To divide one algebraic expression by another invert the divisor and change the operation to multiplication.
After the problem is changed from a division problem to a multiplication problem, it is completed as in the previous section.
FINDING THE LEAST COMMON DENOMINATOROBJECTIVESUpon completing this section you should be able to:
The rule for addition and subtraction of fractions requires that the fractions to be combined must have the same denominator. As preparation for performing these operations we will now investigate the method of finding the least common denominator for any group of fractions. A common denominator lot two or more fractions is an expression that contains all factors of the denominator of each fraction. A least common denominator contains the minimum number of factors to be a common denominator.
Mental arithmetic will allow you to find the least common denominator for small numbers. If asked to add , it is easy to arrive at a least common denominator of 12. If asked how we arrived at 12, we just know that 12 is the least number divisible by both 4 and 6. However, a more involved method is necessary if the numbers are larger or if the fractions are algebraic fractions. Example 1 Find the least common denominator for Solution This problem would require a considerable amount of guesswork, or testing possibilities, if we had no general method.
Let's consider the definition. From it we know that a common denominator for these numbers must contain all factors of each. In other words, we are looking for the smallest number divisible by 12, 14, 15, and 18. The number we are looking for must contain (2)(2)(3) in order to be divisible by 12. It must contain (2)(7) in order to be divisible by 14, and so on. Proceed as follows:
The preceding discussion gives rise to a rule for obtaining a least common denominator for any number of fractions, whether they be numbers or algebraic expressions. To find the least common denominator for two or more fractions:
By inspection of the second denominator we need the additional factor (x  2). The least common denominator is (3x  4)(2x + l)(x  2).
Solution EQUIVALENT FRACTIONSOBJECTIVESUpon completing this section you should be able to:
In further preparation for adding and subtracting fractions, we must be able to change a given fraction to one with a new denominator without changing the value of the original fraction. is called the fundamental principle of fractions. When we analyze this statement, we see two equivalent fractions and note that the numerator and denominator of have both been multiplied by the same nonzero number, a. To change a fraction to an equivalent fraction multiply numerator and denominator by the same nonzero expression.
Solution Since the new denominator is in factored form, by inspection we see that the original denominator (2x + 3) has been multiplied by the factor (x  4). Therefore, the original numerator (x + 1) must also be multiplied by the factor (x  4), giving
Solution Since the original denominator (x  3) has been multiplied by (2) and (x + 1), the original numerator (2x + 1) must also be multiplied by (2) and (x + 1).
ADDITION OF ALGEBRAIC FRACTIONSOBJECTIVESUpon completing this section you should be able to:
We are now prepared to add algebraic fractions by using the techniques discussed in the preceding two sections. You should recall the following rule from arithmetic. The sum of two or more fractions which have the same denominator is the sum of the numerators over their common denominator. Take note that this rule only allows the sum of fractions that have the same denominator. In other words, two or more fractions can only be added if they have a common denominator. The rule for adding any two or more fractions will require the skills developed in the last two sections in addition to knowledge of combining like terms. To add two or more fractions follow these steps:
This answer is in reduced form.
The sum is
We can use fewer written steps if we note that "common denominator" means all the fractions have the same denominator, and if all have the same denominator, then it is necessary to write the denominator only once. To illustrate this we will rework the preceding example.
SUBTRACTION OF ALGEBRAIC FRACTIONSOBJECTIVESUpon completing this section you should be able to:
Subtraction is defined in terms of addition, so the method of subtracting algebraic fractions will be the same as adding algebraic fractions discussed in the preceding section. You will soon see why we have presented them separately. The difference of any two fractions having the same denominator is the difference of their numerators over their common denominator. Notice that this rule is the same as the rule for adding two fractions with the same denominator. The steps for subtracting fractions are, therefore, the same as for adding fractions. To subtract fractions: The obvious question is, "If these two operations are the same, why study them separately?" The answer is that the subtraction gives rise to a very common error that the student must be prepared to avoid.
The error referred to is often made by not recognizing that the minus sign affects the entire numerator of the second fraction and NOT just the first term.
The arrow points out the error most commonly made in subtraction of fractions. The best way to avoid this is to always use parentheses and you are not so likely to fail to change the sign properly.
COMPLEX FRACTIONSOBJECTIVESUpon completing this section you should be able to:
Fractions are defined as the indicated quotient of two expressions. In this section we will present a method for simplifying fractions in which the numerator or denominator or both are themselves composed of fractions. Such fractions are called complex fractions. Thus if the numerator and denominator of a complex fraction are composed of single fractions, it can be simplified by dividing the numerator by the denominator. A generally more efficient method of simplifying a complex fraction involves using the fundamental principle of fractions. We multiply both numerator and denominator by the common denominator of all individual fractions in the complex fraction.
We will use the fundamental principle to again simplify The LCD of 3 and 4 is 12. Thus
EQUATIONS HAVING ALGEBRAIC FRACTIONSOBJECTIVESUpon completing this section you should be able to:
In chapter 2 we encountered equations that have fractions. However, those fractions all had numerical denominators. Now we will discuss equations that have fractions involving variables in the denominators. The method of solving these equations will follow the same pattern as in chapter 2, but there are some additional cautions that you must be prepared to take.
To refresh your memory the steps for solving such equations are repeated here. The main difference in solving equations with arithmetic fractions and those with algebraic fractions comes in checking. The checking process will not just be to find a possible error, but will also be to determine if the equation has an answer. This last possibility arises because with algebraic fractions we multiply by an unknown quantity. This unknown quantity could actually be zero, which would make all the work invalid.
Since division by zero is not possible, we must conclude that x = 1 is not a solution. And since we made no error in the computations we must conclude that this equation has no solution.
Thus, x = 5 is a solution. Therefore, 11 is the amount by which the numerator was increased. SUMMARYKey Words
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