## Interactive Algebra Review

(Based on the algebra reviews in Calculus Applied to the Real World, , and Finite Mathematics and Calculus Applied to the Real World and )

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## 6. Rational Expressions

What is a Rational Expression?

A rational expression is an algebraic expression of the form P/Q, where P and Q are simpler expressions (usually polynomials), and the denominator Q is not zero.

Examples

 (a) 1x - 1 P = 1,   Q = x - 1 (b) 2xy - y22x2 - 1 P = 2xy - y2,   Q = 2x2 - 1 (c) xy2 - y P = Q = (d) y2 + yx - 2 P = Q =

We can manipulate rational expressions in the same way that we manipulate fractions. Here are the basic rules.

Algebra of Rational Expressions

Rule Example
 Multiplication: PQ RS = PRQS
 x + 1x (x - 1)2x + 1 = (x - 1)(x + 1)x(2x + 1) = x2 - 1x(2x + 1)
 Addition with Common Denominator: PQ + RQ = P + RQ
 yxy + 1 + x - 1xy + 1 = x + y - 1xy + 1
 General Addition Rule:(works with or without common denominator) PQ + RS = PS + RQQS
 yx + x - 1y = y2 + x(x - 1)xy
 Subtraction with Common Denominator: PQ - RQ = P - RQ
 yx2 - 1 - x - 1x2 - 1 = -x + y +1x2 - 1
 General Subtraction Rule:(works with or without common denominator) PQ - RS = PS - RQQS
 y2x - xyy + 1 = y2(y + 1) - x2y(x + 1)(y + 1)
Reciprocals:
1
 PQ
=

 QP

1
 x + 1 y - 1
=

 y - 1 x + 1
 Cancellation: PRQR = PQ
 y2(xy - 1)x(xy - 1) = y2x

Rewite each expression as a single rational expression (that is, a ratio of polynomials). Do not factor the numerator or denominator in your answer.

 2x - 3x - 2 . x + 3x + 1
=

 2x - 3x - 2 + x + 3x + 1
=

 x2 - 1x - 2 - 1x - 1
=

2

 x - 2x2
-
1

x - 2
=

 y2x (2x - 3)y + xy
=

 1x + y - 1 x3

y
=

Now go over the examples and try some of the exercises in Section 6 of the Algebra Review of Calculus Applied to the Real World, or Finite Mathematics and Calculus Applied to the Real World.

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Last Updated: January, 1999