1. Rational Expressions and Functions: Adding and Subtracting
Objectives:
When Denominators are the same
When Denominators are different

2. When Denominators Are the Same
Addition and Subtraction with Like Denominators
Let p, q, and r represent polynomials where q ≠ 0. To add or subtract when denominators are the same, add or subtract the numerators and keep the same denominator 1. 2.

3. Adding and Subtracting Rational Expressions with a Common Denominator
The fractions have the same denominator.
Add term in the numerators, and write the result over the common denominator.
Simplify to lowest terms.
Solution

4. Adding and Subtracting Rational Expressions with a Common Denominator
The fractions have the same denominator.
Subtract the terms in the numerators, and write the result over the common denominator.
Simplify the numerator.
Factor the numerator and denominator to determine if the rational expression can be simplified.
Simplify to lowest terms.

5. Adding and Subtracting Rational Expressions with a Common Denominator
The fractions have the same denominator.
Subtract the terms in the numerators, and write the result over the common denominator.
Simplify the numerator.
Factor the numerator and denominator to determine if the rational expression can be simplified.
Simplify to lowest terms.

6. When Denominators Are Different
Factor the denominators of each rational expression.
Identify the LCD
To find the least common multiple of two or more expressions:
Find the prime factorization of each expression
Form a product that contains each factor the greatest number of times that it occurs in any one prime factorization
Rewrite each rational expression as an equivalent expression with the LCD as its denominator.
Add or subtract the numerators, and write the result over the common denominator.
Simplify to lowest terms.

7. Find the least common multiple of each pair of polynomials
Solution
We write the prime factorizations of
The factors 3, 7, and x must appear in the LCM if 21x is to be a factor of the LCM. The factors 3, x, and x must appear in the LCM if 3x² is to be a factor of the LCM

8. ADDING RATIONAL EXPRESSIONS
To add rational expressions, you must have a common denominator. Factor any denominators to help in determining the lowest common denominator.
simplify
distribute
Reduce common factors 1
This fraction needs (x + 5)
This fraction needs nothing
So the common denominator needs each of these factors.
Solution

9. ADDING RATIONAL EXPRESSIONS
To add rational expressions, you must have a common denominator. Factor any denominators to help in determining the lowest common denominator.
Simplify
Add the terms in the numerators, and write the result over the common denominator.
The expressions d - 7 and 7- d are opposites and differ by a factor of -1 Therefore, multiply the numerator and denominator of either expression by -1 to obtain a common denominator.
Solution

10. SUBTRACTING RATIONAL EXPRESSIONS
To subtract rational expressions, you must have a common denominator. Factor any denominators to help in determining the lowest common denominator.
distribute
simplify
This fraction needs (2)
This fraction needs (3)
So the common denominator needs each of these factors.The LCD is 6.
Reduce common factors
NONE
Solution

11. SUBTRACTING RATIONAL EXPRESSIONS
Subtracting rational expressions is much like adding, you must have a common denominator. The important thing to remember is that you must subtract each term of the second rational function.
FOIL
Distribute the negative to each term.
This fraction needs (x + 2)
This fraction needs (x + 6))
So a common denominator needs each of these factors.

12. Find simplified form for the function given by and list all restrictions on the domain
Solutions
Factoring, Note that x ≠ -2, 2
Multiplying by -1/-1 since 2-x is the opposite of x-2

13. The LCD is (x-2)(x+2)
Multiplying by 1 to get the LCD
Removing a factor equal to 1:
that x ≠ -2