1. Basic Algebraic Operations
University of Palestine
IT-College
College Algebra
Basic Algebraic Operations
(Appendix A)
L:3

2. Objectives Cover the topics in Appendix A (A 4):
A-4 : Rational Expressions : basic operations.
Simplifying Rational Expressions
Adding and Subtracting Rational Expressions
Appendix A

3. A-4 : Rational Expressions: basic operations
A-4 : Learning Objectives
After completing this Section, you should be able to:
Find the domain of a rational expression.
Simplify a rational expression.
Find the least common denominator of rational expressions.
Add and subtract rational expressions.
A-4

4. where P and Q are polynomials and Q does not equal 0.
A-4 : Simplifying Rational Expressions
Rational Expression A rational expression is one that  can be written in the form
where P and Q are polynomials and Q does not equal 0.
An example of a rational expression is:
A-4

5. Domain of a Rational Expression
A-4 : Simplifying Rational Expressions
Domain of a Rational Expression
With rational functions, we need to watch out for values that cause our denominator to be 0. 
If our denominator is 0, then we have an undefined value. 
So, when looking for the domain of a given rational function, we use a back door approach. 
We find the values that we cannot use, which would be values that make the denominator 0. 
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6. A-4 : Simplifying Rational Expressions
Example 1:   Find all numbers that must be excluded from the domain of 
Our restriction is that the denominator of a fraction can never be equal to 0. 
So to find what values we need to exclude, think of what value(s) of x, if any, would cause the denominator to be 0. 
*Factor the den.
This give us a better look at it.  Since 1 would make the first factor in the denominator 0, then 1 would have to be excluded.
Since - 4 would make the second factor in the denominator 0, then - 4 would also have to be excluded.
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7. Fundamental Principle of Rational Expressions
A-4 : Simplifying Rational Expressions
Fundamental Principle of  Rational Expressions
For any rational expression  , and any polynomial R, where , , then 
In other words, if you multiply the EXACT SAME thing to the numerator and denominator, then you have an equivalent rational expression.
This will come in handy when we simplify rational expressions, which is coming up next.
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8. Simplifying (or reducing) a Rational Expression
A-4 : Simplifying Rational Expressions
Simplifying (or reducing) a  Rational Expression
Step 1: Factor the numerator and the denominator.  
If you need a review on factoring, by all means go to Factoring Polynomials.  
Step 2: Divide out all common factors that the numerator and the denominator have.
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9. A-4 : Simplifying Rational Expressions
Example 2:  Simplify and find all numbers that must be excluded from the domain of the simplified rational expression:
Step 1: Factor the numerator and the denominator
AND
Step 2: Divide out all common factors that the numerator and the denominator have.
*Factor the trinomials in the num. and den.
*Divide out the common factor of (x + 3)  
*Rational expression simplified
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10. A-4 : Simplifying Rational Expressions
Example 3:  Simplify and find all numbers that must be excluded from the domain of the simplified rational expression
*Factor the diff. of squares in the num. and
  *Factor the trinomial in the den.
*Factor out a -1 from (5 - x)  
*Divide out the common factor of (x – 5)
*Rational expression simplified
A-4

11. Adding or Subtracting Rational Expressions with Common Denominators
A-4 : Adding and Subtracting Rational
Expressions
Adding or Subtracting Rational Expressions  with Common Denominators
Step 1: Combine the numerators together.
Step 2: Put the sum or difference found in step 1 over the common denominator.  
Step 3: Reduce to lowest terms as shown in section: Simplifying Rational Expressions.
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12. A-4 : Adding and Subtracting Rational Expressions
A-4 : Adding and Subtracting Rational
Expressions
Example 1:  Add 
Step 1: Combine the numerators together AND
Step 2: Put the sum or difference found in step 1 over the common denominator.
*Common denominator of 5x - 2  
 *Combine the numerators
*Write over common denominator       
*Excluded values of the original den.
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13. A-4 : Adding and Subtracting Rational Expressions
A-4 : Adding and Subtracting Rational
Expressions
Example 2:  Add 
Step 1: Combine the numerators together AND
Step 2: Put the sum or difference found in step 1 over the common denominator.
*Common denominator of y - 1
*Combine the numerators *Write over common denominator
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14. A-4 : Adding and Subtracting Rational Expressions
A-4 : Adding and Subtracting Rational
Expressions
Step 3: Reduce to lowest terms.
*Factor the num.
*Simplify by div. out the common factor of (y - 1)            
*Excluded values of the original den.
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15. Least Common Denominator (LCD)
A-4 : Adding and Subtracting Rational
Expressions
Least Common Denominator (LCD)
Step 1: Factor all the denominators
 If you need a review on factoring, feel free to go back to factoring Polynomials.
Step 2: The LCD is the list of all the DIFFERENT factors in the denominators raised to the highest power that there is of each factor.
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16. A-4 : Adding and Subtracting Rational Expressions
Adding and Subtracting Rational Expressions  Without a Common Denominator
Step 1: Find the LCD as shown above if needed. Step 2: Write equivalent fractions using the LCD if needed. If we multiply the numerator and denominator by the exact same expression it is the same as multiplying it by the number 1.  If that is the case,  we will have equivalent expressions when we do this. 
Now the question is WHAT do we multiply top and bottom by to get what we want?  We need to have the LCD, so you look to see what factor(s) are missing from the original denominator that is in the LCD.  If there are any missing factors then that is what you need to multiply the numerator AND denominator by.
Step 3: Combine the rational expressions as shown above. Step 4: Reduce to lowest terms as shown in Rational Expressions.
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17. A-4 : Adding and Subtracting Rational Expressions
A-4 : Adding and Subtracting Rational
Expressions
Example 3:  Add 
Step 1: Find the LCD as shown above if needed.   The first denominator has the following two factors:
*Factor the GCF
The second denominator has the following factor:
Putting all the different factors together and using the highest exponent, we get the following LCD:
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18. A-4 : Adding and Subtracting Rational Expressions
A-4 : Adding and Subtracting Rational
Expressions
Step 2: Write equivalent fractions using the LCD if needed.   Since the first rational expression already has the LCD, we do not need to change this fraction
*Rewriting denominator in factored form
Rewriting the second expression with the LCD:
*Missing the factor of (y - 4) in the den.
*Mult. top and bottom by (y - 4)
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19. A-4 : Adding and Subtracting Rational Expressions
A-4 : Adding and Subtracting Rational
Expressions
Step 3: Combine the rational expressions as shown above.
*Combine the numerators *Write over common denominator
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20. A-4 : Adding and Subtracting Rational Expressions
A-4 : Adding and Subtracting Rational
Expressions
Step 4: Reduce to lowest terms. 
*Simplify by div. out the common factor of y    
*Excluded values of the original den.
Note that the values that would be excluded from the domain are 0 and 4.  These are the values that make the original denominator equal to 0.
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21. A-4 : Adding and Subtracting Rational Expressions
A-4 : Adding and Subtracting Rational
Expressions
Example 4:  Add 
Step 1: Find the LCD as shown above if needed. The first denominator has the following factor:
The second denominator has the following two factors:
*Factor the difference of squares
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22. A-4 : Adding and Subtracting Rational Expressions
A-4 : Adding and Subtracting Rational
Expressions
Step 2: Write equivalent fractions using the LCD if needed
  Rewriting the first expression with the LCD:
*Missing the factor of (x + 1) in the den. *Mult. top and bottom by (x + 1)
Since the second rational expression already has the LCD, we do not need to change this fraction.
*Rewriting denominator in factored form
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23. A-4 : Adding and Subtracting Rational Expressions
A-4 : Adding and Subtracting Rational
Expressions
Step 3: Combine the rational expressions as shown above.
*Combine the numerators *Write over common denominator               *Excluded values of the original den.
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24. A-4 : Adding and Subtracting Rational Expressions
A-4 : Adding and Subtracting Rational
Expressions
Step 4: Reduce to lowest terms. 
This rational expression cannot be simplified down any farther. 
  Also note that the values that would be excluded from the domain are
-1 and 1. 
These are the values that make the original denominator equal to 0.
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25. A-4 : Adding and Subtracting Rational Expressions
A-4 : Adding and Subtracting Rational
Expressions
Example 5:  Subtract 
Step 1: Find the LCD as shown above if needed.   The first denominator has the following two factors:
*Factor the trinomial
The second denominator has the following factor:
Putting all the different factors together and using the highest exponent, we get the following LCD:
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26. A-4 : Adding and Subtracting Rational Expressions
A-4 : Adding and Subtracting Rational
Expressions
Step 2: Write equivalent fractions using the LCD if needed.   Rewriting the first expression with the LCD:
*Missing the factor of (x - 8) in the den. *Mult. top and bottom by (x - 8)
Rewriting the second expression with the LCD:
*Missing the factor of (x + 5) in the den. *Mult. top and bottom by (x + 5)
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27. A-4 : Adding and Subtracting Rational Expressions
A-4 : Adding and Subtracting Rational
Expressions
Step 3: Combine the rational expressions as shown above.
*Combine the numerators *Write over common denominator
*Distribute the minus sign through the (   )
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28. A-4 : Adding and Subtracting Rational Expressions
A-4 : Adding and Subtracting Rational
Expressions
Step 4: Reduce to lowest terms. 
*Factor the num.
*No common factors to divide out  
*Excluded values of the original den.
Note that the values that would be excluded from the domain are -5,  -1 and 8.  These are the values that make the original denominator equal to 0.
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29. End of the Lecture
Let Learning Continue