Rational Functions. Do Now Factor the following polynomial completely: 1) x 2 – 11x – 26 2) 2x 3...
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Transcript of Rational Functions. Do Now Factor the following polynomial completely: 1) x 2 – 11x – 26 2) 2x 3...
![Page 1: Rational Functions. Do Now Factor the following polynomial completely: 1) x 2 – 11x – 26 2) 2x 3 – 4x 2 + 2x 3) 2y 5 – 18y 3.](https://reader035.fdocuments.us/reader035/viewer/2022081603/5697c0141a28abf838ccd27c/html5/thumbnails/1.jpg)
Rational Functions
![Page 2: Rational Functions. Do Now Factor the following polynomial completely: 1) x 2 – 11x – 26 2) 2x 3 – 4x 2 + 2x 3) 2y 5 – 18y 3.](https://reader035.fdocuments.us/reader035/viewer/2022081603/5697c0141a28abf838ccd27c/html5/thumbnails/2.jpg)
Do Now
Factor the following polynomial completely:
1) x2 – 11x – 26
2) 2x3 – 4x2 + 2x
3) 2y5 – 18y3
![Page 3: Rational Functions. Do Now Factor the following polynomial completely: 1) x 2 – 11x – 26 2) 2x 3 – 4x 2 + 2x 3) 2y 5 – 18y 3.](https://reader035.fdocuments.us/reader035/viewer/2022081603/5697c0141a28abf838ccd27c/html5/thumbnails/3.jpg)
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.
![Page 4: Rational Functions. Do Now Factor the following polynomial completely: 1) x 2 – 11x – 26 2) 2x 3 – 4x 2 + 2x 3) 2y 5 – 18y 3.](https://reader035.fdocuments.us/reader035/viewer/2022081603/5697c0141a28abf838ccd27c/html5/thumbnails/4.jpg)
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:
![Page 5: Rational Functions. Do Now Factor the following polynomial completely: 1) x 2 – 11x – 26 2) 2x 3 – 4x 2 + 2x 3) 2y 5 – 18y 3.](https://reader035.fdocuments.us/reader035/viewer/2022081603/5697c0141a28abf838ccd27c/html5/thumbnails/5.jpg)
Simplifying Rational Expressions
What if we have an example where we do not have any obvious common factors?
Example 1:
![Page 6: Rational Functions. Do Now Factor the following polynomial completely: 1) x 2 – 11x – 26 2) 2x 3 – 4x 2 + 2x 3) 2y 5 – 18y 3.](https://reader035.fdocuments.us/reader035/viewer/2022081603/5697c0141a28abf838ccd27c/html5/thumbnails/6.jpg)
Simplifying Rational Expressions0 Example 2:
0 Example 3:
0 Example 4:
![Page 7: Rational Functions. Do Now Factor the following polynomial completely: 1) x 2 – 11x – 26 2) 2x 3 – 4x 2 + 2x 3) 2y 5 – 18y 3.](https://reader035.fdocuments.us/reader035/viewer/2022081603/5697c0141a28abf838ccd27c/html5/thumbnails/7.jpg)
Simplifying Rational Expressions
Classwork – Worksheet
Homework – page 577 #3 – 17
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Do Now
Simplify the following rational expressions:
1.
2.
3.
![Page 9: Rational Functions. Do Now Factor the following polynomial completely: 1) x 2 – 11x – 26 2) 2x 3 – 4x 2 + 2x 3) 2y 5 – 18y 3.](https://reader035.fdocuments.us/reader035/viewer/2022081603/5697c0141a28abf838ccd27c/html5/thumbnails/9.jpg)
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:
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Multiplying Rational Expressions0 Example 1: Perform the indicated operation.
0 Example 2: Perform the indicated operation.
![Page 11: Rational Functions. Do Now Factor the following polynomial completely: 1) x 2 – 11x – 26 2) 2x 3 – 4x 2 + 2x 3) 2y 5 – 18y 3.](https://reader035.fdocuments.us/reader035/viewer/2022081603/5697c0141a28abf838ccd27c/html5/thumbnails/11.jpg)
Multiplying Rational Expressions0 Example 3: Perform the indicated operation.
0 Example 4: Perform the indicated operation.
![Page 12: Rational Functions. Do Now Factor the following polynomial completely: 1) x 2 – 11x – 26 2) 2x 3 – 4x 2 + 2x 3) 2y 5 – 18y 3.](https://reader035.fdocuments.us/reader035/viewer/2022081603/5697c0141a28abf838ccd27c/html5/thumbnails/12.jpg)
Multiplying Rational Expressions0 Classwork – Worksheet on Multiplying Rational
Expresssions
0 Homework – Textbook page 578 #18-20, 24-33
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Do Now0 Perform the indicated operation. Be sure your answer is in
simplest form. 1.
2.
3.
![Page 14: Rational Functions. Do Now Factor the following polynomial completely: 1) x 2 – 11x – 26 2) 2x 3 – 4x 2 + 2x 3) 2y 5 – 18y 3.](https://reader035.fdocuments.us/reader035/viewer/2022081603/5697c0141a28abf838ccd27c/html5/thumbnails/14.jpg)
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
![Page 15: Rational Functions. Do Now Factor the following polynomial completely: 1) x 2 – 11x – 26 2) 2x 3 – 4x 2 + 2x 3) 2y 5 – 18y 3.](https://reader035.fdocuments.us/reader035/viewer/2022081603/5697c0141a28abf838ccd27c/html5/thumbnails/15.jpg)
Dividing Rational Expressions0 Example 1: Perform the indicated operation and state the
excluded values
![Page 16: Rational Functions. Do Now Factor the following polynomial completely: 1) x 2 – 11x – 26 2) 2x 3 – 4x 2 + 2x 3) 2y 5 – 18y 3.](https://reader035.fdocuments.us/reader035/viewer/2022081603/5697c0141a28abf838ccd27c/html5/thumbnails/16.jpg)
Example 2: Perform the indicated operation and state the excluded values
![Page 17: Rational Functions. Do Now Factor the following polynomial completely: 1) x 2 – 11x – 26 2) 2x 3 – 4x 2 + 2x 3) 2y 5 – 18y 3.](https://reader035.fdocuments.us/reader035/viewer/2022081603/5697c0141a28abf838ccd27c/html5/thumbnails/17.jpg)
Example 3: Perform the indicated operation and state the excluded values
![Page 18: Rational Functions. Do Now Factor the following polynomial completely: 1) x 2 – 11x – 26 2) 2x 3 – 4x 2 + 2x 3) 2y 5 – 18y 3.](https://reader035.fdocuments.us/reader035/viewer/2022081603/5697c0141a28abf838ccd27c/html5/thumbnails/18.jpg)
Example 4: Perform the indicated operation and state the excluded values
![Page 19: Rational Functions. Do Now Factor the following polynomial completely: 1) x 2 – 11x – 26 2) 2x 3 – 4x 2 + 2x 3) 2y 5 – 18y 3.](https://reader035.fdocuments.us/reader035/viewer/2022081603/5697c0141a28abf838ccd27c/html5/thumbnails/19.jpg)
Example 5: Perform the indicated operation and state the excluded values
![Page 20: Rational Functions. Do Now Factor the following polynomial completely: 1) x 2 – 11x – 26 2) 2x 3 – 4x 2 + 2x 3) 2y 5 – 18y 3.](https://reader035.fdocuments.us/reader035/viewer/2022081603/5697c0141a28abf838ccd27c/html5/thumbnails/20.jpg)
Dividing Rational Expressions0 Class work: Work in partners on Dividing Rational
Expressions Worksheet
0 Homework: Textbook page 578 #34-43
![Page 21: Rational Functions. Do Now Factor the following polynomial completely: 1) x 2 – 11x – 26 2) 2x 3 – 4x 2 + 2x 3) 2y 5 – 18y 3.](https://reader035.fdocuments.us/reader035/viewer/2022081603/5697c0141a28abf838ccd27c/html5/thumbnails/21.jpg)
Do Now
0 Perform the indicated operation:1.
2.
3.
![Page 22: Rational Functions. Do Now Factor the following polynomial completely: 1) x 2 – 11x – 26 2) 2x 3 – 4x 2 + 2x 3) 2y 5 – 18y 3.](https://reader035.fdocuments.us/reader035/viewer/2022081603/5697c0141a28abf838ccd27c/html5/thumbnails/22.jpg)
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
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Example:
![Page 24: Rational Functions. Do Now Factor the following polynomial completely: 1) x 2 – 11x – 26 2) 2x 3 – 4x 2 + 2x 3) 2y 5 – 18y 3.](https://reader035.fdocuments.us/reader035/viewer/2022081603/5697c0141a28abf838ccd27c/html5/thumbnails/24.jpg)
Example:
![Page 25: Rational Functions. Do Now Factor the following polynomial completely: 1) x 2 – 11x – 26 2) 2x 3 – 4x 2 + 2x 3) 2y 5 – 18y 3.](https://reader035.fdocuments.us/reader035/viewer/2022081603/5697c0141a28abf838ccd27c/html5/thumbnails/25.jpg)
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
![Page 26: Rational Functions. Do Now Factor the following polynomial completely: 1) x 2 – 11x – 26 2) 2x 3 – 4x 2 + 2x 3) 2y 5 – 18y 3.](https://reader035.fdocuments.us/reader035/viewer/2022081603/5697c0141a28abf838ccd27c/html5/thumbnails/26.jpg)
Example:
![Page 27: Rational Functions. Do Now Factor the following polynomial completely: 1) x 2 – 11x – 26 2) 2x 3 – 4x 2 + 2x 3) 2y 5 – 18y 3.](https://reader035.fdocuments.us/reader035/viewer/2022081603/5697c0141a28abf838ccd27c/html5/thumbnails/27.jpg)
Find the LCDs of the following expressions:
a)
b)
c)
![Page 28: Rational Functions. Do Now Factor the following polynomial completely: 1) x 2 – 11x – 26 2) 2x 3 – 4x 2 + 2x 3) 2y 5 – 18y 3.](https://reader035.fdocuments.us/reader035/viewer/2022081603/5697c0141a28abf838ccd27c/html5/thumbnails/28.jpg)
DAY 2
Example:
Find the LCD: xy
Multiply by what’s missing
Simplify!
![Page 29: Rational Functions. Do Now Factor the following polynomial completely: 1) x 2 – 11x – 26 2) 2x 3 – 4x 2 + 2x 3) 2y 5 – 18y 3.](https://reader035.fdocuments.us/reader035/viewer/2022081603/5697c0141a28abf838ccd27c/html5/thumbnails/29.jpg)
Example:
Find the LCD: 90xy2
Multiply by what’s missing
Simplify!
![Page 30: Rational Functions. Do Now Factor the following polynomial completely: 1) x 2 – 11x – 26 2) 2x 3 – 4x 2 + 2x 3) 2y 5 – 18y 3.](https://reader035.fdocuments.us/reader035/viewer/2022081603/5697c0141a28abf838ccd27c/html5/thumbnails/30.jpg)
Example:
Find the LCD:
-What can’t x equal??
Multiply by what’s missing
Simplify!
![Page 31: Rational Functions. Do Now Factor the following polynomial completely: 1) x 2 – 11x – 26 2) 2x 3 – 4x 2 + 2x 3) 2y 5 – 18y 3.](https://reader035.fdocuments.us/reader035/viewer/2022081603/5697c0141a28abf838ccd27c/html5/thumbnails/31.jpg)
Example:
-LCD?
-What can’t x
equal??
![Page 32: Rational Functions. Do Now Factor the following polynomial completely: 1) x 2 – 11x – 26 2) 2x 3 – 4x 2 + 2x 3) 2y 5 – 18y 3.](https://reader035.fdocuments.us/reader035/viewer/2022081603/5697c0141a28abf838ccd27c/html5/thumbnails/32.jpg)
Solving Rational EquationsExample:
**this is a proportion, can solve using cross multiplication
5 (x – 1) = 2 (15)
5x – 5 = 30
5x = 35
x = 7
![Page 33: Rational Functions. Do Now Factor the following polynomial completely: 1) x 2 – 11x – 26 2) 2x 3 – 4x 2 + 2x 3) 2y 5 – 18y 3.](https://reader035.fdocuments.us/reader035/viewer/2022081603/5697c0141a28abf838ccd27c/html5/thumbnails/33.jpg)
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
![Page 34: Rational Functions. Do Now Factor the following polynomial completely: 1) x 2 – 11x – 26 2) 2x 3 – 4x 2 + 2x 3) 2y 5 – 18y 3.](https://reader035.fdocuments.us/reader035/viewer/2022081603/5697c0141a28abf838ccd27c/html5/thumbnails/34.jpg)
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
![Page 35: Rational Functions. Do Now Factor the following polynomial completely: 1) x 2 – 11x – 26 2) 2x 3 – 4x 2 + 2x 3) 2y 5 – 18y 3.](https://reader035.fdocuments.us/reader035/viewer/2022081603/5697c0141a28abf838ccd27c/html5/thumbnails/35.jpg)
Solve:
![Page 36: Rational Functions. Do Now Factor the following polynomial completely: 1) x 2 – 11x – 26 2) 2x 3 – 4x 2 + 2x 3) 2y 5 – 18y 3.](https://reader035.fdocuments.us/reader035/viewer/2022081603/5697c0141a28abf838ccd27c/html5/thumbnails/36.jpg)
Solve:
![Page 37: Rational Functions. Do Now Factor the following polynomial completely: 1) x 2 – 11x – 26 2) 2x 3 – 4x 2 + 2x 3) 2y 5 – 18y 3.](https://reader035.fdocuments.us/reader035/viewer/2022081603/5697c0141a28abf838ccd27c/html5/thumbnails/37.jpg)
Solve: