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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:
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