1. Section R5: Rational Expressions
223 Reference Chapter
Section R5: Rational Expressions
The quotient of two polynomials is called a Rational Expression.
Determining the Domain: Remember, the denominator cannot be zero (that would make the fraction undefined.) So a rational expression would have a domain of all real numbers, with the exception of values that make the denominator zero.
Example: Find the domain of each of the following expressions 1. 2. 3.

2. Section R5: Rational Expressions
223 Reference Chapter
Section R5: Rational Expressions
The quotient of two polynomials is called a Rational Expression.
Determining the Domain: Remember, the denominator cannot be zero (that would make the fraction undefined.) So a rational expression would have a domain of all real numbers, with the exception of values that make the denominator zero.
Example: Find the domain of each of the following expressions 1.
all real numbers except -4
All real numbers except -2, 3
All real numbers except -3, -7/2

3. Section R5: Rational Expressions
223 Reference Chapter
Section R5: Rational Expressions
Simplifying Rational Expressions.
Write each numerator and denominator in terms of its factors, then simplify.
Example: Write each expression in lowest terms. 1. 2. 3.

4. Section R5: Rational Expressions
223 Reference Chapter
Section R5: Rational Expressions
Simplifying Rational Expressions.
Write each numerator and denominator in terms of its factors, then simplify.
Example: Write each expression in lowest terms. 1.
1/5
2m + 6
r + 1
r + 3

5. Section R5: Rational Expressions
223 Reference Chapter
Section R5: Rational Expressions
Simplifying Rational Expressions involving Multiplication.
Write each numerator and denominator in terms of its factors, cancel out common factors, and simplify.
Example: Find each product. 1. 2. 3.

6. Section R5: Rational Expressions
223 Reference Chapter
Section R5: Rational Expressions
Simplifying Rational Expressions involving Multiplication.
Write each numerator and denominator in terms of its factors, cancel out common factors, and simplify.
Example: Find each product. 1.
2a + 8
c + 4
c - 4 x 4

7. Section R5: Rational Expressions
223 Reference Chapter
Section R5: Rational Expressions
Simplifying Rational Expressions involving Division.
First, change division to multiplication by finding the reciprocal of the second rational expression; then, write each numerator and denominator in terms of its factors, cancel out common factors, and simplify.
Example: Find each product. 1. 2. 3.

8. Section R5: Rational Expressions
223 Reference Chapter
Section R5: Rational Expressions
Simplifying Rational Expressions involving Division.
First, change division to multiplication by finding the reciprocal of the second rational expression; then, write each numerator and denominator in terms of its factors, cancel out common factors, and simplify.
Example: Find each product. 1.
2b - 6
b + 1
d^2 + 3d + 9 3

9. Section R5: Rational Expressions
223 Reference Chapter
Section R5: Rational Expressions
Simplifying Rational Expressions involving Addition and/or Subtraction.
First, factor all of the numerators and denominators and determine the Least Common Denominator (LCD). Use the LCD to make both fractions have the same denominator, then add and/or subtract the fractions and simplify.
Example: Find each sum or difference. 1. 2.

10. Section R5: Rational Expressions
223 Reference Chapter
Section R5: Rational Expressions
Simplifying Rational Expressions involving Addition and/or Subtraction.
First, factor all of the numerators and denominators and determine the Least Common Denominator (LCD). Use the LCD to make both fractions have the same denominator, then add and/or subtract the fractions and simplify.
Example: Find each sum or difference.
x^2 – 2x
x^2 - 16
y^2 – 2y - 2
6y^2

11. Section R5: Rational Expressions
223 Reference Chapter
Section R5: Rational Expressions
Simplifying Rational Expressions involving Addition and/or Subtraction.
Example: Find each sum or difference. 3. 4.

12. Section R5: Rational Expressions
223 Reference Chapter
Section R5: Rational Expressions
Simplifying Rational Expressions involving Addition and/or Subtraction.
Example: Find each sum or difference.
m + 17
m^3 – m^2 –
a^2 + 3a + 5
a^3 - a

13. Section R5: Rational Expressions
223 Reference Chapter
Section R5: Rational Expressions
Simplifying Complex Fractions.
The quotient of two rational expressions is referred to as a Complex Fraction. To simplify, multiply both the numerator and denominator by the LCD of all of the fractions.
Example: Simplify the expression.

14. Section R5: Rational Expressions
223 Reference Chapter
Section R5: Rational Expressions
Simplifying Complex Fractions.
The quotient of two rational expressions is referred to as a Complex Fraction. To simplify, multiply both the numerator and denominator by the LCD of all of the fractions.
Example: Simplify the expression.
5c + 8
c - 4

15. Section R5: Rational Expressions
223 Reference Chapter
Section R5: Rational Expressions
Simplifying Complex Fractions.
Example: Simplify the expression.

16. Section R5: Rational Expressions
223 Reference Chapter
Section R5: Rational Expressions
Simplifying Complex Fractions.
Example: Simplify the expression.
6k - 5
k + 2