Interactive Algebra Review

Based on the algebra reviews in Applied Calculus and Finite Mathematics and Applied Calculus

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4. Rational Algebraic Expressions

Note You need to understand how to multiply algebraic expressions using the distributive law before starting work on this tutorial. If you feel you need to review this, go back to 3. Multiplying and Factoring Algebraic Expressions.

Q What is a Rational Expression?

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

 1x - 1
P = 1,   Q = x - 1
 2xy - y22x2 - 1
P = 2xy - y2,   Q = 2x2 - 1
 x2 + 3x - 4
P = x2 + 3x - 4,   Q = 1

Here are some for you.

Q1
 y2 + yx - 2
is rational with P =
Q =

Q2  xy2 - y is rational with P =
Q =

Q3  (x2 - y2)-1 is rational with P =
Q =

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

Algebra of Rational Expression

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) - x2yx(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

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

Q1
 2x - 3x - 2 . x + 3x + 1
=

Q2
 2x - 3x - 2 + x + 3x + 1
=

Q3
 x2 - 1x - 2 - 1x - 1
=

Q4
2
 x - 2x2
-
1

x - 2
=

Q5
 y2x (2x - 3)y + xy
=

Q6
 1x + y - 1 x3

y
=

Now go over the examples and try some of the exercises in Section A.4 of the Algebra Review in Applied Calculus and Finite Mathematics and Applied Calculus. Alternatively, go on to the next tutorial (Solving Equations).

Last Updated: March, 2006