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Absolute Values
Solving Two-Step Equations Algebraically
Multiplying Monomials
Factoring Trinomials
Solving Quadratic Equations
Power Functions and Transformations
Composition of Functions
Rational Inequalities
Equations of Lines
Graphing Logarithmic Functions
Elimination Using Multiplication
Multiplying Large Numbers
Multiplying by 11
Graphing Absolute Value Inequalities
Polynomials
The Discriminant
Reducing Numerical Fractions to Simplest Form
Addition of Algebraic Fractions
Graphing Inequalities in Two Variables
Adding and Subtracting Rational Expressions with Unlike Denominators
Multiplying Binomials
Graphing Linear Inequalities
Properties of Numbers and Definitions
Factoring Trinomials
Relatively Prime Numbers
Point
Inequalities
Rotating a Hyperbola
Writing Algebraic Expressions
Quadratic and Power Inequalities
Solving Quadratic Equations by Completing the Square
BEDMAS & Fractions
Solving Absolute Value Equations
Writing Linear Equations in Slope-Intercept Form
Adding and Subtracting Rational Expressions with Different Denominators
Reducing Rational Expressions
Solving Absolute Value Equations
Equations of a Line - Slope-intercept form
Adding and Subtracting Rational Expressions with Unlike Denominators
Solving Equations with a Fractional Exponent
Simple Trinomials as Products of Binomials
Equivalent Fractions
Multiplying Polynomials
Slope
Graphing Equations in Three Variables
Properties of Exponents
Graphing Linear Inequalities
Solving Cubic Equations by Factoring
Adding and Subtracting Fractions
Multiplying Whole Numbers
Straight Lines
Solving Absolute Value Equations
Solving Nonlinear Equations
Factoring Polynomials by Finding the Greatest Common Factor
Logarithms
Algebraic Expressions Containing Radicals 1
Addition Property of Equality
Three special types of lines
Quadratic Inequalities That Cannot Be Factored
Adding and Subtracting Fractions
Coordinate System
Solving Equations
Factoring Polynomials
Solving Quadratic Equations
Multiplying Radical Expressions
Solving Quadratic Equations Using the Square Root Property
The Slope of a Line
Square Roots
Adding Polynomials
Arithmetic with Positive and Negative Numbers
Solving Equations
Powers and Roots of Complex Numbers
Adding, Subtracting and Finding Least Common Denominators
What the Factored Form of a Quadratic can tell you about the graph
Plotting a Point
Solving Equations with Variables on Each Side
Finding the GCF of a Set of Monomials
Completing the Square
Solving Equations with Radicals and Exponents
Solving Systems of Equations By Substitution
Adding and Subtracting Rational Expressions
Percents
Laws of Exponents and Dividing Monomials
Factoring Special Quadratic Polynomials
Radicals
Solving Quadratic Equations by Completing the Square
Reducing Numerical Fractions to Simplest Form
Factoring Trinomials
Writing Decimals as Fractions
Using the Rules of Exponents
Evaluating the Quadratic Formula
Rationalizing the Denominator
Multiplication by 429
Writing Linear Equations in Point-Slope Form
Multiplying Radicals
Dividing Polynomials by Monomials
Factoring Trinomials
Introduction to Fractions
Square Roots
   
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Adding and Subtracting Rational Expressions

Definition A rational expression is a ratio (fraction) of two polynomials. All rational expressions are valid for all real numbers except those values that make the denominator zero.

are rational expressions whereas are not.

 

When adding (or subtracting) rational expressions with a common denominator, the process is straightforward: Simply add (or subtract) the numerators, and put the result in a rational expression with the same denominator; i.e.,

where a, b and c are polynomials.

Example 1

Example 2

Note that there is no restriction on x because , as will be shown in another paper.

When performing any operation with rational expressions of a single variable, we must make sure our final answer is simplified:

a. All polynomials are in descending order; i.e., the largest power of the variable appears on the left and as we proceed to the right, the powers of the variable decrease.

b. All leading terms (highest degree terms) in the denominator have positive coefficients.

c. All like (similar) terms have been combined.

d. The greatest common factor of the numerator and denominator is 1; i.e., all common factors have been divided out.

e. Unless told otherwise, the numerator is to multiplied out and simplified, while the denominator may be left in factored form.

The expressions, , are considered to be simplified. The expressions, are not considered to be simplified. The following shows how to convert the latter pair of expressions into simplified forms.

[Factor out -1 from denominator]

[Put in descending order]

[Factor out -1 from (-x + 1)]

[Eliminate the leading neg sign in denom.]

[Simplify]

Example 3

Substract:

Solution: The factorizations of the denominators are

Hence the LCD is:

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

Use these factorizations to convert the rational expressions to rational expressions that have the LCD for their denominators, perform the subtraction, and simplify the result.

[Acceptable as being simplified]

[Preferrable]

Note that the above is valid for all . What this means is that if we substitute any value of

into , and evaluate the expressions, we obtain the same result.

For example, let x = 3. Then we obtain

and

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