Math 3201
4.2 Simplifying Rational Expressions
Simplifying rational expressions is similar to simplifying algebraic fractions.
Example 1:
(A)
3
6
(B)
4
(C)
6
25
15
Simplifying a rational expression results in an expression that is easier to work with than
the original.
For example, suppose we given the rational expression
𝑥 2 +4𝑥
where 𝑥 = 0. The simplified
𝑥
form of this expression is 𝑥 + 4, where 𝑥 ≠ 0. Suppose we are asked to evaluate the
rational expression for 𝑥 = 2.
(A) Which form of the equation would you prefer to work with? Explain.
(B) What is the benefit of simplifying a rational expression before substituting values
in?
(C) Why does the simplified form of the expression (𝑥 + 4) include the restriction
𝑥 ≠ 0, even though this form of the expression is not written as a fraction?
Before we can simplify rational expressions we much briefly review a couple of types of
factoring that we learned in Math 1201/2201.
Review of Factoring
Removing a GCF


Find a GCF. Includes the GCF of the numbers and the lowest power of any common
variable.
Factor out the GCF. What you will have left in the brackets is what you
get when you divide each term by the GCF.
Example 1:
Factor by removing a GCF.
(A) 5𝑥 2 + 15𝑥
(B) 18𝑦 4 − 16𝑦 3
Difference of Squares
This can be used when the following conditions are met:
 You have two terms present.
 Both terms are perfect squares.
 The two terms are separated by a subtraction sign.
In general, 𝑎2 − 𝑏 2 = (𝑎 + 𝑏)(𝑎 − 𝑏)
Example 2:
Factor each difference of squares.
(A) 36𝑥 2 − 49𝑦 2
(B) 81𝑦 4 − 16
Steps for Simplifying Rational Expressions
 Completely factor the numerator and denominator.
 Cancel any factors that are common to both.
 Find the non-permissible values for the original rational expression.
 Use the non-permissible values to state restrictions for the rational expression.
Example 3:
Simplify the following and state restrictions.
(A)
(C)
5𝑥
3𝑥
6𝑥+3
8𝑥+4
(B)
(D)
6𝑥 2
3𝑥 5
5𝑥+20
15𝑥−5
(E)
(G)
6𝑥 3 +27𝑥 2
4𝑥 2 +18𝑥
3𝑥 2 −12
6𝑥+12
(F)
(H)
𝑥 2 −9
2𝑥 2 −6𝑥
5𝑥 2 −405
10𝑥+90
(I)
16𝑥 3 −12𝑥 2
32𝑥 3 −18𝑥
(J)
4𝑥 2 −6𝑥
16𝑥 4 −81
Example 4:
Identify and correct the errors in the simplification of rational expression.
(A)
(B)
(C)
Textbook Questions: page 229 - 231 #1, 2, 3, 4, 5, 6, 7, 8, 9, 13