Name: Date: Period: Math Stretch: Factor the following expressions completely: a) 𝑥! − 16 b) 𝑥! + 2𝑥

c) 𝑥! − 4𝑥 − 12 c) 3𝑥! + 10𝑥 − 8

Notes: Simplifying Rational Expressions A rational expression is in simplest form when: Let a, b and c be nonzero real numbers or variable expressions. Then the following property applies:

To simplify a rational expression: Example 1: Simplify the expression below completely:

Example 2: Simplify the expression below completely:

𝑦! − 3𝑦! − 18𝑦2𝑦! + 5𝑦 + 3

Simplifying Practice:Simplify the expression, if possible. State the domain of the function (excluded values).

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2) !!!!"!!!!!!!!!!!

3) !!!!!!!!!!

4) !!!!!!!!!!!!"

5) (!!!)(!!!)(!!!)(!!!)

6) !!!!!!!!"!!!

7) Write a rational expression that has a domain of 𝑥 ≠ 3 and simplifies to 𝑥 + 5:

Notes: Multiplying & Dividing Rational Expressions To multiply rational expressions, multiply the __________________________ and then multiply the ___________________________. Simplify if possible.

Example:

To divide rational expressions. Multiply the first expression by the _________________________ of the second expression.

Example:

Multiplying and Dividing Practice:

Multiply or divide the expression and simplify completely:

1) !!!

!!∙ !!

!!!!!

2) !!!!!!!!!!!!!!

∙ !!!!!!!"!!!!

3) !!!!!!!!!!

÷ !!!!!!!!!!!

4) 𝑥! + 9𝑥 + 18÷ !!!!!!!"!!!!!!!"

Notes: Adding & Subtracting Rational Expressions To add (or subtract) rational expressions with like denominators, simply add (or subtract) their ________________________. Then place the result over the common denominator.

Example with like denominators:

To add (or subtract) two rational expressions with UNLIKE denominators: Find the ______________ ___________________ _______________________ (LCD), which is the

_______________ __________________ ___________________ (LCM) of the denominators.

Example 1 with unlike denominators:

𝟐𝒂+

𝟑𝒂− 𝟓

Example 2 with unlike denominators:

𝟕𝟑𝒙− 𝟗−

𝒙+ 𝟒𝒙𝟐 − 𝟗

Example 3 with unlike denominators:

What is the sum of 𝟑

𝒙!𝟑 and

𝒙𝟑!𝒙

Adding and Subtracting Practice: Add or subtract the expression and simplify completely: 1) !

!!!!+ !

!!!! 2) !

!!!!!!!+ !

!!!

3) !!!!!

− !!!!!"

4) !!!!!!

− !!!!!

5) !!!!"!"!!!

+ !!!!

6) !!!!!!!!!!

+ !!!!!!!!!!"

+ !!!!!!!!