REDUCING FRACTIONS TO
LOWEST TERMS
There are many useful applications of factoring. One of the most important is that of
simplifying algebraic fractions. Fractions that contain algebraic expressions in the numerator
or denominator, or both, can be reduced to lower terms, if there are factors common to
numerator and denominator. If the terms of a fraction are monomials, common factors are
immediately apparent, as in the following expression:
If the terms of a fraction are polynomials, the polynomials must be factored in order to
recognize the existence of common factors, as in the following two examples:
Notice that without the valuable process of factoring, we would be forced to use the fractions
in their more complicated form. When there
are factors common to both numerator and denominator, it is obviously more practical to
cancel them (first using the factoring process) before proceeding.
Practice problems. Reduce to lowest terms in each of the following:
OPERATIONS INVOLVING FRACTIONS
Addition, subtraction, multiplication, and division operations involving algebraic fractions
are often simplified by means of factoring, whereas they would be quite complicated with
out the use of factoring.
MULTIPLYING FRACTIONS
Multiplication of fractions that contain polynomials is similar to
multiplication of fractions that contain only
arithmetic numbers. If this fact is kept in mind, the
student will have little difficulty in mastering
multiplication in algebra. For instance, we recall
that to multiply a fraction by a whole number, we simply multiply the numerator
by the whole number. This ,is illustrated in the following example:
Arithmetic:
Algebra:
Sometimes the work may be simplified by factoring and canceling before
carrying out this multiplication. The following example illustrates this:
When the multiplier is a fraction, the ruler of arithmetic
remain applicablethat is, multiply numerators
together and denominators together, This is
illustrated ,as follows:
Arithmetic:
Algebra:
Where possible, the work may be considerably reduced
by factoring, canceling, and then carrying out the multiplication, as in the
following example:
Although the factors may be multiplied to form two
trinomials as shown, it is usually sufficient to leave
the answer in factored form. Practice problems. In the
following problems, multiply as indicated:
Answers:
