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)
    1

    x - 1
          P = 1,   Q = x - 1
    (b)
    2xy - y2

    2x2 - 1
    P = 2xy - y2,   Q = 2x2 - 1
    (c) xy2 - y P =
    Q =
    (d)
    y2 + y

    x - 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:
P

Q
R

S
=
PR

QS
 
x + 1

x
(x - 1)

2x + 1
=
(x - 1)(x + 1)

x(2x + 1)
=
x2 - 1

x(2x + 1)
Addition with Common Denominator:
P

Q
+
R

Q
=
P + R

Q
 
y

xy + 1
+
x - 1

xy + 1
=
x + y - 1

xy + 1
General Addition Rule:
(works with or without common denominator)
P

Q
+
R

S
=
PS + RQ

QS
 
y

x
+
x - 1

y
=
y2 + x(x - 1)

xy
Subtraction with Common Denominator:
P

Q
-
R

Q
=
P - R

Q
 
y

x2 - 1
-
x - 1

x2 - 1
=
-x + y +1

x2 - 1
General Subtraction Rule:
(works with or without common denominator)
P

Q
-
R

S
=
PS - RQ

QS
 
y2

x
-
xy

y + 1
=
y2(y + 1) - x2y

(x + 1)(y + 1)
Reciprocals:
1
P

Q
=
 
Q

P
 
 
1
x + 1

y - 1
=
 
 
y - 1

x + 1
 
 
Cancellation:
PR

QR
=
P

Q
 
y2(xy - 1)

x(xy - 1)
=
y2

x

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 - 3

x - 2
.
x + 3

x + 1
=


2x - 3

x - 2
+
x + 3

x + 1
=


x2 - 1

x - 2
-
1

x - 1
=


2

x - 2

x2
-
1

x - 2
=


y2

x
(2x - 3)

y
+
x

y
=


1

x + 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.

Alternatively, press "next" button on the sidebar to go on to the next topic.

Last Updated: January, 1999
Copyright © 1998 Stefan Waner and Steven R. Costenoble