Simplifying Rational Expressions
Canceling Like Factors
When we reduce a common fraction such as
we do so by noticing that there is a factor common to both the numerator and the denominator (a factor of 2 in this example), which we can divide out of both the numerator and the denominator.
We use exactly the same procedure to reduce rational expressions.
Polynomial / Monomial
Each term in the numerator must have a factor that cancels a common factor in the denominator.
but
cannot be reduced because the 2 is not a common factor of the entire numerator.
WARNING You can only cancel a factor of the entire numerator with a factor of the entire denominator
However, as an alternative, a fraction with more than one term in the numerator can be split up into separate fractions with each term over the same denominator; then each separate fraction can bereduced if possible:
· Think of this as the reverse of adding fractions over a common denominator. Sometimes this is a useful thing to do, depending on the circumstances.You end up with simpler fractions, but the price you pay is that you have more fractions than you started with. 
 Polynomials must be factored first. You can’t cancel factors unless you can see the factors:
Example:
 Notice how canceling the (x– 2) from the denominator left behind a factor of 1
Multiplication and Division
Same rules as for rational numbers!
Multiplication
 Both the numerators and the denominators multiply together
 Common factors may be cancelled before multiplying
Example:
Given Equation:  
First factor all the expressions: (I also put the denominators in parentheses because then it is easier t see them as distinct factors) 

Now cancel common factors—any factor on the top can cancel with any factor on the bottom:


Now just multiply what’s left. You usually do not have to multiply outthe factors, just leave them as shown. 
Division
 Multiply by the reciprocal of the divisor
 Invert the second fraction, then proceed with multiplication as above
 Do not attempt to cancel factors before it is written as a multiplication
Addition and Subtraction
Same procedure as for rational numbers!
 Only the numerators can be addedtogether, and only when all the denominators are the same
Finding the LCD
 The LCD is built up of all the factors of the individual denominators, each factor included the most number of times it appears in an individual denominator.
 The product of all the denominators is always a commondenominator, but not necessarily the LCD (the final answer may have to be reduced).
Example:
Given equation:  
Factor both denominators:  
Assemble the LCD: Note that the LCD contains bothdenominators 

Build up the fractions so that they both have the LCD for a denominator: (keep both denominators in factored form to make it easier to see what factors they need to look like the LCD) 

Now that they are over the same denominator, you can add the numerators:  
And simplify: 
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