Exponential Functions
Linera Equations
Simple Trinomials as Products of Binomials
Laws of Exponents and Dividing Monomials
Solving Equations
Multiplying Polynomials
Multiplying and Dividing Rational Expressions
Solving Systems of Linear Inequalities
Mixed-Number Notation
Linear Equations and Inequalities in One Variable
The Quadratic Formula
Fractions and Decimals
Graphing Logarithmic Functions
Multiplication by 111
Solving Systems of Equations - Two Lines
Solving Nonlinear Equations by Factoring
Solving Linear Systems of Equations by Elimination
Rationalizing the Denominator
Simplifying Complex Fractions
Factoring Trinomials
Linear Relations and Functions
Axis of Symmetry and Vertices
Equations Quadratic in Form
The Appearance of a Polynomial Equation
Subtracting Reverses
Non-Linear Equations
Exponents and Order of Operations
Factoring Trinomials by Grouping
Factoring Trinomials of the Type ax 2 + bx + c
The Distance Formula
Invariants Under Rotation
Multiplying and Dividing Monomials
Solving a System of Three Linear Equations by Elimination
Multiplication by 25
Powers of i
Solving Quadratic and Polynomial Equations
Slope-intercept Form for the Equation of a Line
Equations of Lines
Square Roots
Integral Exponents
Product Rule for Radicals
Solving Compound Linear Inequalities
Axis of Symmetry and Vertices
Multiplying Rational Expressions
Reducing Rational Expressions
Properties of Negative Exponents
Numbers, Factors, and Reducing Fractions to Lowest Terms
Solving Quadratic Equations
Factoring Completely General Quadratic Trinomials
Solving a Formula for a Given Variable
Factoring Polynomials
Decimal Numbers and Fractions
Multiplication Properties of Exponents
Multiplying Fractions
Multiplication by 50

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We multiply two rational numbers by multiplying their numerators and multiplying their denominators. For example,

Instead of reducing the rational number after multiplying, it is often easier to reduce before multiplying. We first factor all terms, then divide out the common factors, then multiply:

When we multiply rational numbers, we use the following definition.


Multiplication of Rational Numbers

If and are rational numbers, then


We multiply rational expressions in the same way that we multiply rational numbers: Factor all polynomials, divide out the common factors, then multiply the remaining factors.


Example 1

Multiplying rational expressions

Find each product of rational expressions.


a) First factor the coefficients in each numerator and denominator:

  Divide out the common factors.
  Quotient rule


Do not attempt to divide out the x in . This expression cannot be reduced because x is not a factor of both terms in the denominator. Compare this expression to the following:

In Example 2(a) we will multiply a rational expression and a polynomial. For Example 2(b) we will use the rule for factoring the difference of two cubes.


Example 2

Multiplying rational expressions

Find each product.


a) First factor the polynomials completely:

  Divide out the common factors.

b) Note that a - b is a factor of a3 - b3 and b - a occurs in the denominator. We can factor b - a as -1(a - b) to get a common factor: