Rational Expressions
Math 3 Notes Day 1
Simplifying and Multiplying Rational Expressions
I. Extraneous Roots
A rational expression can be evaluated just as any polynomial, except that a rational expression can be undefined when the denominator is equal to zero. In Algebra we called this finding the extraneous roots.
Identify the extraneous roots for each of the following.
Examples:
1. [pic] 2. [pic] 3. [pic]
II. Simplifying Rational Expressions
Concept Example: 15 =
35
Steps to Simplifying a Rational Expression
1) Factor the numerator and the denominator completely
2) Divide common factors
Examples:
Simplify.
1. [pic] 2. [pic]
3. [pic] 4. [pic]
5. [pic] 6. [pic]
Your Turn (
Example: Simplify
a) x3 + 8 b) x4 ( y4
x2 ( 3x ( 10 x2 ( y2
III. Multiplying Rational Expressions
If all things are considered equal, multiplication of rational expressions is just the same as simplifying, except in the end we have to write the answer as one consolidated expression.
Steps for Multiplying Rational Expressions
1) Factor numerators and denominators completely
2) Divide common factors if possible
3) Multiply numerators and denominators (Be flexible with this)
Examples:
Simplify.
1. [pic] 2. [pic]
3. [pic] 4. [pic]
Your Turn (
Example: Multiply
a. [pic] b. [pic]
Math 3 Notes Day 2
Dividing Rational Expressions
Case 1: Dividing Two Polynomials by Simplifying
1. [pic] 2. [pic]
Case 2: Dividing Two Polynomials by Long Division
3. [pic] 4. (12x3 + 2 + 11x + 20x2) ÷ (2x + 1)
Case 3: Dividing Two Polynomials by Synthetic Division
5. Divide [pic] by [pic] 6. (3x2 – 13x – 10) ÷ (x + 5)
Case 4: Dividing Two Rational Expressions (This is the one we will focus on today.)
Since division is just multiplication by the reciprocal, nothing changes from multiplication.
Steps for Dividing Rational Expressions
1) Take the reciprocal of the divisor.
2) Multiply the dividend and the reciprocal of the divisor
Examples:
7. [pic] 8 . [pic]
9. [pic] 10. [pic]
Your Turn (
Divide.
a. [pic] b. [pic]
Case 5: Dividing Complex Fractions
A complex fraction is a fraction with an expression in the numerator and an expression in the denominator.
11. [pic] 12. [pic]
Your turn (
a. . [pic]
Math 3 Notes Day 3
Addition and Subtraction of Rational Expressions
Addition/Subtraction of Rational Expressions with Like Denominators
When we have a common denominator, as with fractions, we simply add or subtract the numerators. We do have to be cautious because when subtracting it is the entire numerator that's subtracted, so we must use the distributive property to subtract.
Concept Examples:
2 + 6 = 2x + 5 ( x ( 5 =
5 5 y + 2 y + 2
Addition/Subtraction of Rational Expressions with Unlike Denominators
1) Figure out what each denominator needs in order for all denominators to be the same
2) Multiply the numerator by what was missing on each fraction
3) Add or Subtract as normal
4) Keep the denominator
Concept Example: [pic]
Simplify.
1. [pic] 2. [pic]
3. [pic] 4. [pic]
5. [pic] 6. [pic]
7. [pic] 8. [pic]
|Simplifying Complex Fractions Method #1 (Division) |Simplifying Complex Fractions Method #2 |
| |(mult. by LCD) |
|1) Solve or simplify the problem in the numerator |1) Find the LCD of the numerator and the denominator fractions |
|2) Solve or simplify the problem in the denominator |2) Multiply numerator and denominator by LCD |
|3) Divide the numerator by the denominator |3) Simplify resulting fraction |
|4) Reduce | |
|[pic] |[pic] |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
Solve using the method of your choice.
5. [pic] 6. [pic]
7. [pic] 8. [pic]
Math 3 Notes Day 4
Solving Rational Equations
The only difference between solving rational expression equations and regular equations is that all solutions must be checked to make sure that it does not make the denominator of the original expression zero. If one of the solutions makes the denominator zero, it is called an extraneous solution.
Solving a Rational Expression Equation
Step 1: Identify the extraneous roots of the rational expressions
Step 2: Find the LCD of the denominators in the equation
Step 3: Multiply all terms by the LCD
Step 4: Solve appropriately (may be either a linear or a quadratic, so be careful!)
Step 5: Eliminate extraneous solutions as possible solutions to the equation and write as
a solution set.
Examples:
A. [pic] B. [pic]
Extraneous Roots:_______________ Extraneous Roots:_______________
C. [pic] D. [pic]
Extraneous Roots:_______________ Extraneous Roots:_______________
E. [pic] F. [pic]
Extraneous Roots:_______________ Extraneous Roots:_______________
G. [pic] H. [pic]
Extraneous Roots:_______________ Extraneous Roots:____
Your turn (
a. [pic]
Extraneous Roots:_______________
Math 3 Notes Day 5
Solving Radical Equations
An equation that has a radical with a variable under the radicand is called a radical equation. The radicand is the numbers and variables under the [pic] symbol. That is the number that we are taking the square root of. For example, the radicand in[pic] is 3x.
Steps for Solving Radical Equations with One Radical
1. Isolate the radical.
2. Square BOTH sides. The most common mistake is students only square one side.
3. Isolate the variable.
4. Check your answer – it is possible that your answer will not work (no solution).
Example 1
1. [pic] 2. [pic]
3. [pic] 4. [pic]
5. [pic]
Steps for Solving Radical Equations with Two Radicals
1. Separate the radicals so that one is on each side of the equation
2. Square BOTH sides. The most common mistake is students only square one side. Note: If radical is not over the entire side of the equation, you will have to square the binomial. Ex: (x+3)2 = x2 +6x+9
3. Isolate the variable.
4. Check your answer – it is possible that your answer will not work (no solution).
Examples:
1. [pic] 2. [pic]
3. [pic] 4. [pic]
Math 3 Notes Day 6
Rational Equations: Applications and Problem Solving
Formula for “work related” problems:
|Rate |Time |Work Completed |
|(always a fraction 1/rate) |(always the time it takes together) |(always equal to 1) |
|[pic] | | |
|[pic] | | |
1. Huck Finn can paint a fence in 5 hours. After some practice, Tom Sawyer can now paint the fence in 6 hours. How long would it take Huck and Tom to paint the fence together?
Practice:
Mr. Falcigno has discovered that a large dog can destroy his entire garden in 2 hours and that a small boy can do the same job in 1 hour. How long would it take the large dog and the small boy working together to destroy Mr. Falcigno’s garden?
2. Every week, Cindy must stuff 1000 envelopes. She can do the job by herself in 6 hours. If Laura helps, they get the job done in 5.5 hours. How long would it take Laura to do the job by herself?
Practice:
Bob can paint a fence in 5 hours, and working with Jen, the two of them painted the fence in 2 hours. How long would it have taken Jen to paint the fence alone?
3. If one inlet pipe can fill the pool in 2.5 hours and a second outlet pipe can drain the pool in 2 hours, how long does it take to fill the pool if both pipes are working together?
Practice:
An inlet pipe can fill a tank in 9 hours. It takes 12 hours for the drain pipe to empty the tank. How long wil it take to fill the tank if both the inlet and the drain pipe are open?
4. It takes Jane 4 hours to complete her math homework. Alice can complete the assignment in 3 hours. If Jane works ½ hour before Alice joins her, how long will it take the two of them to finish the homework?
Practice:
George takes 15 hours to do a job that his father can do in 6 hours. If george works for 4.5 hours before his father joins him, how long will it take the two of them working together to finish the job?
5. Anne and Maria play tennis almost about every weekend. So far, Anne has won [pic] out of [pic] matches.
a. How many matches will Anne have to win in a row to improve her winning percentage to [pic]?
b. How many matches will Anne have to win in a row to improve her winning percentage to [pic]?
c. Can Anne reach a winning percentage of [pic]?
d. After Anne has reached a winning percentage of [pic] by winning consecutive matches as in part (b), how many matches can she now lose in a row to have a winning percentage of [pic]?
Practice
You have [pic] liters of a juice blend that is [pic] juice.
a. How many liters of pure juice need to be added in order to make a blend that is [pic] juice?
b. How many liters of pure juice need to be added in order to make a blend that is [pic] juice?
c. Suppose that you have added [pic] liters of juice to the original [pic] liters. What is the percentage of juice in this blend?
d. Solve your equation in part (c) for the amount [pic]. Are there any excluded values of the variable [pic]? Does this make sense in the context of the problem?
6. The sum of a number and its reciprocal is 13/6. Find the number.
Practice
The sum of a number and its reciprocal is 53/14. Find the number.
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