Perform Rational Expression Operations

Lesson 7 of 7: Arithmetic with Polynomials (+)

In this lesson:

• Multiply, divide, add, and subtract rational expressions
• Use the rational number analogy to justify each operation
• State and maintain domain restrictions throughout

Learning Objectives for This Lesson

By the end of this lesson, you should be able to:

1. Multiply and divide rational expressions, canceling common polynomial factors
2. Add and subtract rational expressions with the same denominator
3. Add and subtract rational expressions with different denominators using the LCD
4. State and apply domain restrictions: exclude all where any denominator equals zero
5. Explain each operation via the rational number analogy

Recall: The Four Fraction Arithmetic Rules

• Multiply: — multiply tops, multiply bottoms
• Divide: — flip and multiply
• Add/Subtract: find LCD, build equivalent fractions, combine numerators

What is ? What LCD do you need?

Fraction Rules Already in Your Toolkit

— multiply tops, multiply bottoms

— flip the divisor

— common denominator

Every one of these rules works the same way for polynomial fractions.

From Long Division to Full Arithmetic

APR.D.6 converted a single rational expression to form.

Today: treat rational expressions as objects and perform all four operations on them.

Same objects, new operations: , , , with .

The Rational Number Analogy: Fraction Rules Apply

Every rational number rule has a rational expression counterpart.

Learn the structure once — apply it to both integers and polynomials.

Domain Restrictions: Exclude Division by Zero

A rational expression is undefined wherever .

State restrictions before simplifying.

Example: — factor: ; domain: ,

After simplification, the restrictions remain even if the factor canceled.

Check-In: Find Domain of Each Expression

State the domain for each:

Factor denominators, set each factor equal to zero, state restrictions.

Multiply and Divide: Factor First, Then Cancel

Multiply: factor all numerators and denominators; cancel common factors; multiply.

Divide: rewrite as multiplication by the reciprocal; then factor and cancel.

Rule: cancel factors of the entire numerator/denominator — never cancel additive terms.

Worked: Factor First, Then Multiply

Factor: ,

Cancel and : ,

Worked: Flip and Multiply to Divide

Rewrite: . Factor: ;

Cancel , , : ,

Probe: Can You Cancel the ?

Is ?

Test at :

No. The in is a term, not a factor — it cannot be canceled.

Only factors of the entire numerator or denominator can cancel.

Why Addition Needs a Common Denominator

— denominators are different; you must find the LCD first.

Same principle for rational expressions:

Find the LCD: . Build equivalent fractions. Then add.

Building the LCD from Factored Denominators

How to find the LCD: factor denominators; take each distinct factor to its highest power.

• and : both factored → LCD
• and : → LCD

Worked: Add Fractions with Same Denominator

Add numerators, keep the denominator:

Factor: . Domain: .

Worked: Find LCD and Add Unlike Fractions

LCD . Build equivalent fractions:

Worked: Subtract with a Quadratic Denominator

Factor: . LCD .

Probe: Domain Restriction After Cancellation

Canceling gives — but what happens at ?

The original is undefined at . The simplified form is not. The domain restriction must be stated.

Check-In: Subtract and State Domain Restrictions

Compute . State domain restrictions.

Find the LCD, build equivalent fractions, subtract numerators, state restrictions.

Complex Problems: All Four Skills Together

Factor; simplify first term: . LCD .

Three Critical Errors to Prevent

Canceling terms: — is a term, not a factor

Dropping restrictions: only for

Sign error in subtraction: means subtract all of — put it in parentheses

Summary: Rational Expression Arithmetic Rules

Four operations, same rules as fractions:

• Multiply: factor, cancel, multiply remaining factors
• Divide: flip second fraction; factor, cancel, multiply
• Add/subtract (same denominator): combine numerators
• Add/subtract (different denominators): factor; find LCD; build equivalents; combine

State domain restrictions from the original; they survive all operations.

Next: Rational Equations and Partial Fractions

Rational equations (REI.A.2): set a rational expression equal to zero or another; clear denominators; solve; check for extraneous solutions.

Partial fractions (calculus): the reverse of addition — decompose back into .

Everything in this lesson feeds both directions.