Math Review of Division and Reciprocals of Rational Expressions

Overview

Division of rational expressions is similar to division of real numbers. In order to divide by a rational expression, the rational expression is changed to its reciprocal, then multiplied.

Reciprocals

The product of two reciprocals is 1, whether the expression is a number, a monomial with variables, or a polynomial. For example, the reciprocal of the rational expression 4/5 is 5/4, because 4/5 ∙ 5/4 is 1. The reciprocal of the monomial 3m² is 1/(3m²), because 3m² times 1/(3m²) is 1. Similarly, the reciprocal of the polynomial (2x² – 3)/(x + 4) is (x + 4)/(2x² – 3).

Figure 1: The product of reciprocals is 1.

Division of Monomials

In order to divide a rational expression, multiply by the reciprocal, similar to dividing by fractions, and then simplify. Suppose 4/5 is divided by 2/3. In order to divide, 4/5 is multiplied by the reciprocal of 2/3 or 3/2. 4/5 times 3/2 is 12/10, which can be simplified to 6/5. Similarly, suppose that (6x)/5 is divided by (2x)/10. The reciprocal of (2x)/10 is 10/(2x). 6x/5 ∙ 10/2x or (6x ∙ 10)/(5 ∙ 2x), which can be further simplified to 60x/10x or 6.

Figure 2: To divide a rational expression, multiply by the reciprocal and simplify.

Division of Rational Expressions

Suppose that (x + 1)/(x + 2) is divided by (x – 1)/(x + 3). The expression (x + 1)/(x + 2) is multiplied by the reciprocal or (x + 3)/(x – 1). The result, [(x + 1)(x + 3)]/[(x + 2)(x – 1)] is already in simplest form. Suppose that x equals 2. Then ¾ would be divided by 1/5, or multiplied by 5. The improper fraction 15/4 is already in simplest terms.

Division of Polynomials

Suppose that (x + 1)/(x² – 1) is divided by (x + 1)/(x² -2x + 1). The expression (x + 1)/(x² -1) is multiplied by the reciprocal (x² – 2x + 1)/ (x + 1). The expression (x + 1)/(x + 1) can be cancelled out, to leave (x² -2x + 1)/(x² – 1). Neither expression is in simplest terms, because (x² -2x + 1) is the perfect square trinomial (x – 1)(x – 1) and (x² – 1) is the difference of squares, (x + 1)(x – 1). The expression (x – 1)/(x – 1) can be cancelled out to leave (x-1)/(x +1).

Figure 3: The steps for the division of polynomials.

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