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Simplifying Rational Expressions

The steps for simplifying in this section are to factor then cancel . Given a rational
expression, the quotient of two polynomials , we will factor the numerator and
denominator if we can and cancel factors that are exactly the same.

Rational Expression – The quotient of two polynomials where .

When evaluating rational expressions we can plug in the appropriate values either before
we simplify or after , the result will be the same. Although, it is more efficient to simplify
first then evaluate.

Evaluate when x=-1 Evaluate when x=-1  

Now evaluate when x=-1.

 
Factor the numerator and denominator first then cancel.
 

We can see that when evaluating, the result will be the same whether or not we simplify
first. It turns out that not all numbers can be used when we evaluate.

Evaluate when   To find restrictions
This indicates that we
have a restriction, or
that

The point is that not all real numbers will be defined in the above rational expression. In
fact there are two restrictions to the domain, x≠− 2 and . These values , when
plugged in, will result in zero in the denominator . Another way to say this is that the
domain consists of all real numbers except for −2 and. Tip: To find the restrictions,
set each factor in the denominator “not equal” to zero and solve . The factors in the
numerator do not contribute to the list of restrictions.

A. Simplify and state the restrictions to the domain.

Even if the factor cancels it still contributes to the list of restrictions. Basically, it is
important to remember the domain of the original expression when simplifying. Also, we
must use caution when simplifying, please do not try to take
obviously incorrect short cuts like this :

 

will never
equal zero so there
are no real restrictions

Since subtraction is not commutative we must be alert to factors that are opposites. For
example notice that 5 – 3 = 2 and 3 – 5 = – 2. In general .

Opposite Binomial Property :

Alternatively,

B. Simplify and state the restrictions to the domain.

 
Since addition
is commutative,
x+3 = 3+x, these
factors cancel.

 

At this point we will do the same type of problems using function notation.

C. Simplify the function and evaluate.

Given find Given find
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