Rational Functions

Do Now

Factor the following polynomial completely:

1) x2 – 11x – 26

2) 2x3 – 4x2 + 2x

3) 2y5 – 18y3

Rational FunctionsA RATIONAL FUNCTION has the form where p(x) and q(x) are polynomials and q(x) ≠ 0.

A rational expression is in SIMPLIFIED FORM if its numerator and denominator have no common factors other than 1.

Simplifying Rational Expressions

To simplify a rational expression: Let a, b, and c be expressions with b ≠ 0 and c ≠ 0. Then the following property applies:

Ex:

Simplifying Rational Expressions

What if we have an example where we do not have any obvious common factors?

Example 1:

Simplifying Rational Expressions0 Example 2:

0 Example 3:

0 Example 4:

Simplifying Rational Expressions

Classwork – Worksheet

Homework – page 577 #3 – 17

Do Now

Simplify the following rational expressions:

1.

2.

3.

Multiplying Fractions0 How do we multiply fractions? 0 Example:

0 The rule for multiplying rational expressions is the same as multiplying numerical fractions:0 Multiply numerators0 Multiply denominators0 Write new fraction in simplified form.

0 Property:

Multiplying Rational Expressions0 Example 1: Perform the indicated operation.

0 Example 2: Perform the indicated operation.

Multiplying Rational Expressions0 Example 3: Perform the indicated operation.

0 Example 4: Perform the indicated operation.

Multiplying Rational Expressions0 Classwork – Worksheet on Multiplying Rational

Expresssions

0 Homework – Textbook page 578 #18-20, 24-33

Do Now0 Perform the indicated operation. Be sure your answer is in

simplest form. 1.

2.

3.

Dividing Rational Expressions0 How do we divide fractions?

Example:

0 RULE:0 Keep – change – flip(keep the first fraction the same, change division to multiplication, flip the second fraction)0 Multiply as usual0 Property:

0 Excluded values: values that make the denominator equal to zero

Dividing Rational Expressions0 Example 1: Perform the indicated operation and state the

excluded values

Example 2: Perform the indicated operation and state the excluded values

Example 3: Perform the indicated operation and state the excluded values

Example 4: Perform the indicated operation and state the excluded values

Example 5: Perform the indicated operation and state the excluded values

Dividing Rational Expressions0 Class work: Work in partners on Dividing Rational

Expressions Worksheet

0 Homework: Textbook page 578 #34-43

Do Now

0 Perform the indicated operation:1.

2.

3.

Adding and Subtracting Rational Expressions

-Combine the numerators together

-Put the sum or difference in step 1 over the common denominator

-Reduce to lowest terms

Example:

Example:

What if we do NOT have an LCD???

Step 1: Find LCD

Step 2: Multiply the numerator(s)

by the factor that is missing

Step 3: Combine and simplify as

usual

Example:

Find the LCDs of the following expressions:

a)

b)

c)

DAY 2

Example:

Find the LCD: xy

Multiply by what’s missing

Simplify!

Example:

Find the LCD: 90xy2

Multiply by what’s missing

Simplify!

Example:

Find the LCD:

-What can’t x equal??

Multiply by what’s missing

Simplify!

Example:

-LCD?

-What can’t x

equal??

Solving Rational EquationsExample:

**this is a proportion, can solve using cross multiplication

5 (x – 1) = 2 (15)

5x – 5 = 30

5x = 35

x = 7

Example: Solve the equation

Step 1: Find LCD and state the excluded values

5x ( x + 2 )

x cannot equal 0 or -2

Step 2: Multiply all expressions by LCD to cancel out the denominators

Step 3: Simplify, solve, and check

Solve:

Step 1: Find LCD and state the excluded values

Step 2: Multiply all expressions by LCD to cancel out the denominators

Step 3: Simplify, solve, and check

Solve:

Solve:

Solve: