At first glance the factors might be mistakenly considered as common, or the fraction might be mistakenly considered as already simplified. Solution This type of problem requires special attention because it is a common cause for error. Thus when the factor (2x + 5) is divided, the factor 1 remains. Note that the numerator 2x + 5 can be written as (2x 4- 5) * 1. Here (x + 2) is a common factor, so both numerator and denominator may therefore be divided. Solution Here you may use "trial and error" for the numerator and "grouping" for the denominator. 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. The given fraction is already in simplified form. Notice that even though we were able to factor the numerator and denominator, we still cannot divide since no factors are common to both. Note that this is incorrect since they are terms and not factors. In an expression such as some students are tempted to divide the 3's.
We can only divide common factors, never common terms. Now divide by the common factor (x + 2) in both numerator and denominator to get Remember, a factor divided by itself is equal to 1. Next divide by the common factors, giving The fraction, however, is not in simplified form since numerator and denominator have a common factor of 2. To simplify a fraction factor the numerator and denominator completely and then divide both numerator and denominator by all common factors. To obtain the simplified form of a fraction apply the following rule. In simplifying fractions this way you were using the following definition.Ī fraction is in simplified (or reduced) form if the numerator and denominator contain no common factor (other than 1).Ī fraction such as is in simplified form since the numerator 2 and the denominator 3 have no common factor other than one. The fraction cannot be reduced because no number (other than 1) will divide both numerator and denominator. For the fraction you "reduced" it to by dividing the numerator and the denominator by 4. In your study of arithmetic you were instructed that fractional answers were always to be left in reduced, or simplified form. Factor the numerator and denominator of a fraction.Īn algebraic fraction is the indicated ratio of two algebraic expressions.Upon completing this section you should be able to: SOLVING EQUATIONS INVOLVING SIGNED NUMBERS OBJECTIVES We will now study operations on fractions whose components are algebraic expressions. However, the fractions of arithmetic are made up strictly of numbers. They occur in formulas and in many day-to-day practical problems. You have encountered fractions many times since early in the study of mathematics.
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