1.4 - Rational Expressions

Section 1.4 Rational

Expressions

Chapter 1 - Fundamentals

1.4 - Rational Expressions

DefinitionsFractional Expression

A quotient of two algebraic expressions is called a fractional expression.

Rational ExpressionA rational expression is a fractional expression where both the numerator and denominator are polynomials.

2

3 2

3 4 1

x x

x x x

2

3 2

3 4 1

x x

x x x

1.4 - Rational Expressions

DomainThe domain of an algebraic expression is the set of

real numbers that the variable is permitted to have.

1.4 - Rational Expressions

Basic Expressions & Their Domains

Expression Domain (Set Notation) Domain (Interval Notation)

1

x

x

1

x

| 0x x

| 0x x

| 0x x

,0 0,

[0, )

0,

1.4 - Rational Expressions

Simplifying Rational Expressions

To simplify rational expressions we must

1. Factor the numerator and denominator completely.2. State the restrictions or domain.3. Reduce the common factors from the numerator and

denominator.

1.4 - Rational Expressions

Example 1Simplify the following expression.

3 2

3

2 3

4

x x x

x x

1.4 - Rational Expressions

Multiplying Rational Expressions

To multiply rational expressions we must

1. Factor the numerator and denominator completely.2. State the restrictions or domain.3. Multiple factors.4. Reduce the common factors from the numerator and

denominator.

1.4 - Rational Expressions

Example 2Perform the multiplication and simplify.

2 2

3 2 3 12

3 4 6 13 6

x x

x x x x

1.4 - Rational Expressions

Dividing Rational Expressions

To divide rational expressions we must

1. Factor the numerator and denominator completely.2. State the restrictions or domain.3. Invert the divisor and multiply.4. State new restrictions.5. Reduce the common factors from the numerator and

denominator.

1.4 - Rational Expressions

Example 3 – pg. 42 #33Perform the multiplication and simplify

2 2

2 2

2 3 1 6 5

2 15 2 7 3

x x x x

x x x x

1.4 - Rational Expressions

Adding or Subtracting Rational Expressions

To add or subtract rational expressions we must

1. Factor the numerator and denominator completely.2. State the restrictions or domain.3. Find the LCD.4. Combine fractions using the LCD.5. Use the distributive property in the numerator and

combine like terms. 6. If possible, factor the numerator and reduce

common terms.

1.4 - Rational Expressions

Example 4 – pg. 42 #48Perform the addition or subtraction and simplify.

2 2

2 3 4

a ab b

1.4 - Rational Expressions

Compound FractionsA compound fraction is a fraction in which the

numerator, denominator, or both, are themselves fractional expressions.

1

1

xyxy

1.4 - Rational Expressions

Simplifying Compound Fractions

To simplify compound expressions we must

1. Factor the numerator and denominator completely.2. State the restrictions or domain.3. Find the LCD.4. Multiply the numerator and denominator by the

LCD to obtain a fraction.5. Simplify.6. If possible, factor.

1.4 - Rational Expressions

Example 5 – pg. 42 #60Perform the addition or subtraction and simplify.

11

11

11

c

c

1.4 - Rational Expressions

RationalizingIf a fraction has a numerator (or denominator) in the

form then we may rationalize the numerator (or denominator) by multiplyting both the numerator and denominator by the conjugate radical .

A B C

A B C

1.4 - Rational Expressions

Example 6 – pg. 43 #81

Rationalize the denominator.

1

2 3

1.4 - Rational Expressions

Example 7 – pg. 43 #87

Rationalize the numerator.

1 5

3