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Rational expressions

An expression that is the quotient of two algebraic expressions (with denominator not 0) is called a fractional expression.  The most common fractional expressions are those that are the quotients of two polynomials; these are called rational expressions.  Since fractional expressions involve quotients, it is important to keep track of values of the variable that satisfy the requirement that no denominator be0. For example, x != -2 in the rational expression:

because replacing x with -2 makes the denominator equal 0.  Similarly, in


x!=-2 and x!= -4


The restrictions on the variable are found by determining the values that make the denominator equal to zero. In the second example above, finding the values of x that make (x + 2)(x + 4) = 0 requires using the property that ab = 0 if and only if a = 0 or b = 0,  as follows.


x+2=0 or x+4=0

x=-2 or x=-4

Just as the fraction 6/8 is written in lowest terms as 3/4, rational expressions may also be written in lowest terms. This is done with the fundamental principle.

fundamental principle of fractions

Example 1

Write each expression in lowest terms.

rational expression with polynomial in numerator and denominator

Factor the numerator and denominator to get

factoring before reducing

By the fundamental  principle,

result of reducing

In the original expression p cannot be 0 or -4, because

denominator can not be 0

So this result is valid only for values of p other than 0 and -4.   From now on, we shall always assume such restrictions when reducing rational expressions.

Now let's take a look at how our step-by-step fraction solver solves this problem:


Example 2

another fraction to reduce

Factor to get

factoring a fraction's numerator and denominator

The factors 2 - k and k - 2 have opposite signs.  Because of this, multiply numerator and denominator by -1,   as follows.

multiplying by -1

Since (k-2)*(-1)=-k+2  or 2- k,

factoring numerator and denominator


simplified fraction

Working in an alternative way would lead to the equivalent result

alternate version oof the result

Our fraction calculator can solve this and many similar problems. If you would like a similar problem to be generated, click on solve similar button:


Probably the most common error made in algebra is the incorrect use of the fundamental principle to write a fraction in lowest terms, Remember, the fundamental principle requires a pair of common factors, one in the numerator and one in the denominator. For example,

correct way of reducing

On the other hand

final result

cannot be simplified further by the fundamental principle, because the numerator cannot be factored.