Interactive Algebra Review

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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.

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