

Rational Expressions

Simplification of rational expressions, reducing to lowest
terms

Addition and subtraction of rational
expressions

Multiplication and division of rational
expressions

Solving complex rational expressions






Simplification of rational expressions, reducing to lowest
terms

A
rational expression is a fraction of which the numerator and the
denominator are polynomials. 
A
rational expression is reduced to lowest terms if all common
factors from the numerator and denominator are canceled. 
To
reduce a rational expression to lowest terms first factorize
both the numerator and denominator as much as possible then
cancel common factors, i.e., divide their numerator and
denominator by common factors. 

Examples:
Reduce the following
rational expression to lowest terms. 



factoring
out the minus sign, 

or 







Examples:
Reduce or simplify the following
rational expression to lowest terms. 


Addition and subtraction of rational
expressions

To add or subtract two fractions with
the same denominator, add or subtract the numerators and write the sum over the common denominator. 
To add or subtract fractions with different
denominators: 
First find
the least common denominator (lcd the smallest number that can be
divided by each denominator). 
Write equivalent fractions using this
denominator. Then add or subtract the fractions. 
The
process for rational expressions is identical. 

Examples:
Perform the indicated
operations and reduce the answer to lowest terms. 


Multiplication and division of rational
expressions

Use
the formulas for the multiplication
and the division of fractions. 


and 



Notice
the two special cases of which should be aware of: 


and 




Examples:
Perform the indicated
operations and reduce the answer to lowest terms. 

Note,
since each second line should be subtracted, the sign of each term is reversed. 

Solving complex rational expressions

A complex or compound rational expression has fractional expressions in its numerator, denominator or both. 
To simplify complex fractions, change the complex fraction into a division problem, i.e., divide the numerator by the denominator. 
Examples:
Simplify complex
fractions. 









Intermediate
algebra contents 



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