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Rational ExpressionsSection P.6

Prerequisites

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A rational expression is a ratio of two polynomials in which the denominator is not the zero polynomial.

The domain of a rational expression is the set of all real numbers that can be used in place of the variable.

Note: Rational expressions are to algebra what fractions are to arithmetic.

In arithmetic we learned that each rational number has infinitely many equivalent forms. This fact is due to the basic principle of rational numbers.

Basic Principle of Rational NumbersIf a, b, and c are integers with b 0 and c 0, then

.ba

bcac

Reducing

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Definition: Multiplication of Rational Numbers

If a/b and c/d are rational numbers, then

.bdac

dc

ba

We multiply rational expressions in the same manner as rational numbers.

Of course, any common factor can be divided out as we do when reducing rational expressions.

Multiplication

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We divide rational numbers by multiplying by the reciprocal of the divisor, or invert and multiply.

Rational expressions are divided in the same manner as rational numbers.

Definition: Division of Rational NumbersIf a/b and c/d are rational numbers with c 0, then

.cd

ba

dc

ba

Division

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The addition of fractions can be carried out only when their denominators are identical.

To get a required denominator, we may build up the denominator of a fraction.

We multiply the numerator and denominator of a fraction by the same nonzero number to get an equivalent fraction.

Definition: Addition and Subtraction of Rational Numbers

If a/b and c/b are rational numbers, then

and

bca

bc

ba .

bca

bc

ba

Addition and Subtraction

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For fractions with different denominators, we build up one or both denominators to get denominators that are equal to the least common multiple (LCM) of the denominators.

The least common denominator (LCD) is the smallest number that is a multiple of all of the denominators.

Procedure: Finding the LCD

1. Factor each denominator completely.2. Write a product using each factor that appears in a

denominator. 3. For each factor, use the highest power of that factor that

occurs in the denominators.

Addition and Subtraction