Rational Expressions
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Rational Expressions
The quotient of two polynomials is called a rational expression. Rational expressions are to
polynomials what rational numbers are to integers. The goal here is to learn to do everything
that you do with Rational Numbers with rational expressions.
The isn't too difficult as long as you know two things wel , rational number arithmetic and
polynomial factoring.
If you need some review on either of these please read Rational Numbers or Factoring
Polynomials. It is also a good idea to have an understanding of finding LCMs of numbers by
prime factoring, because we will be doing the same thing for polynomials. For review on that
see How to Prime Factor a Number and GCFs and LCMs.
Simplification is also cal ed reducing to lowest terms with number fractions. When you reduce
fractions to lowest terms you divide top and bottom by common factors. See Rational
Numbers for an explanation of why this can be done.
With algebraic fractions you do the same thing, and you must remember that it is only
common factors that you can divide out.
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You normally keep track of what you are doing with this by factoring the top and bottom and
then crossing out any factors that are the same upstairs as downstairs, but make sure you
remember that you are taking out common factors, so you must factor before you cancel
anything out, and you can only cross out common factors of the whole top and bottom, not just
anything that looks the same upstairs and downstairs.
In general in algebra if you are ever unsure about whether you can do something one way you
can test it out is to replace the variables with numbers and see if what you are thinking about
doing would stil make sense.
A "rational expression" is a polynomial fraction, and anything you could do with regular
fractions you can do with rational expressions. However, since there are variables in rational
expressions, there are some additional considerations.
When you dealt with fractions, you knew that the fraction could have any whole numbers for
the numerator and denominator, as long as you didn't try to divide by zero.
When dealing with rational expressions, you wil often need to evaluate the expression, and it
can be useful to know which values would cause division by zero, so you can avoid these x
values. So probably the first thing you'll do with rational expressions is find their domains.
Find the domain of 3/x. : The domain is all values that x is allowed to be. Since I can't
divide by zero (division by zero isn't al owed), I need to find al values of x that would cause
division by zero. The domain wil then be all other xvalues. When is this denominator equal to
zero? When x = 0.
Determine the domain of x/3. : The domain doesn't care what is in the numerator of a
rational expression. The domain is only influenced by the zeroes of the denominator. Will "3"
ever equal zero? Of course not. Since the denominator will never equal zero, no matter what
value x is, then there are no forbidden values for this rational expression, and x can be
anything. So the domain is "al x".
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