1. Factor both the numerator and
denominator of the fraction, if
necessary.
Example 1: What is the reduced form of:
6x  x  4
Example 2: Simplify:
x 2  9 x  14
x2  2x  8
Example 3: Simplify
 x  4   x 2  25 
2. Reduce the fraction.
3. Rewrite any remaining
expressions in the numerator and
denominator.
Simplifying Rational Expressions – Section 8.1
x3 y  6 x3
6x2  x2 y
A rational expression is undefined when the denominator is equal to zero.
1. Set the denominator equal to zero.
Example 1: Under what conditions is the expression
6x  x  4
undefined?
x

 4   x 2  25 
Example 2: Under what conditions is the expression
x 2  9 x  14
undefined?
x2  2 x  8
2. Factor, if necessary.
3. Set each piece equal to zero and solve.
Where Rational Expressions are Undefined – Section 8.1
Example 1: Simplify:
1. Multiply the numerators.
10 x 2
5 y3

4
40 y 100 x5
1. Rewrite the expression by leaving
the first fraction alone, change the
division symbol to multiplication,
and flip the second fraction. (Keep,
change, flip)
Example 2: Simplify:
16 x 2 y 4 4 x3 y

7 st 5
14s 3t10
2. Multiply the numerators.
2. Multiply the denominators.
3. Multiply the denominators.
3. Simplify.
4. Simplify.
Multiplying & Dividing Rational Expressions (monomials) –
Section 8.1
Example 1: Simplify:
x2  7 x  8 x  5

x 8
9x  9
1. Rewrite the expression by
leaving the first fraction alone,
change the division symbol to
multiplication, and flip the
second fraction. (Keep,
change, flip)
Example 2: Simplify:
x 2  16 x 2  14 x  40
 2
x 9
x  x  90
1. Factor the numerator.
2. Factor the numerator.
2. Factor the denominator.
3. Factor the denominator.
3. Reduce the fraction.
4. Reduce the fraction.
4. Rewrite any remaining
expressions in the numerator
and denominator.
5. Rewrite any remaining
expressions in the numerator
and denominator.
Multiplying & Dividing Rational Expressions (polynomials) –
Section 8.1
x3
16 x 2  25 y 2
Example 1: Simplify:
x4
4x  5 y
1. Rewrite the problem as a division problem.
2. Rewrite the expression by leaving the first fraction
alone, change the division symbol to multiplication, and
flip the second fraction. (Keep, change, flip)
3. Factor the numerator.
4. Factor the denominator.
5. Reduce the fraction.
6. Rewrite any remaining expressions in the numerator
and denominator.
Complex Fractions – Section 8.1
Example 1: Find the LCM of
5
7
and 2
.
x  8x
x  9x  8
2
1. Factor each denominator into prime factors.
2. Write the first denominator down.
3. Look at each additional denominator and write down any new factors that
were not in the first denominator. If the same factor appears more than once
in the any of the remaining denominators, then you must write down each
additional occurrence as part of the LCD.
Finding the Least Common Denominator or LCM – Section 8.2
Example 1:
1. Factor the denominator of each fraction to help find the LCD.
6
2
 2
.
x  4 x  5x  6
2
5
3
 2
.
Example 2 2 x  5 x  3 2 x  11x  5
2
2. Find the least common denominator (LCD).
3. Rewrite the factions with the common denominator.
To find the new numerators for each fraction compare the denominator of
each of the original fractions to the LCD and figure out what is missing.
Multiply numerator and denominator by the missing factor(s)
4. Simplify the numerator by distributing and combining like terms.
WHEN SUBTRACTION: IT IS IMPORTANT to place parenthesis
around the second fraction. Be sure to distribute the negative!
5. Combine the fractions by adding the numerators and keeping the LCD.
6. Factor the numerator if you can.
7. Simplify or reduce the rational expression if you can.
Adding/Subtracting Rational Expressions – Section 8.2
a
General form for reciprocal functions: f ( x ) 
k
xh
1. Find the vertical asymptote at x = h. The domain is all real number
EXCEPT h. (REMEMBER: the sign of h is the OPPOSITE of the sign in the
equation)
2. Find the vertical asymptote at y = k. The range is all real number
EXCEPT k. (REMBEMER: the sign of k is the SAME as the sign in the
equation
Example 1: Graph f ( x) 
2
1
x3
Vertical Asymptote
Domain
Horizontal Asymptote
Range
X
3. Graph the asymptotes on the graph with a DOTTED line.
4. Find the vertical asymptote in the table (the value will have an ERROR in
the y column.) Plot the two points above the error in the table, and the two
points below the error.
5. Connect the points with a curve, and DO NOT cross the asymptotes.
Graphing Reciprocal Functions – Section 8.3
Y
ERROR
Vertical Asymptote(s):
Set the denominator equal to 0 and solve.
Example: Find the vertical and horizontal asymptotes.
𝑥 2 −1
𝑦 = 𝑥 2 +3𝑥+2
y
x
x 1
y
x2
x2  3
Horizontal Asymptote:
Compare the degree of the numerator (m) to the degree of the denominator (n).
m < n: y = 0
𝑎
m = n: 𝑦 = , where a is the leading coefficient of the numerator and b
𝑏
is the leading coefficient of the denominator
m > n: does not exist
x – intercept:
Set the numerator equal to 0 and solve.
y – intercept:
Let x = 0 and solve.
2
Example: Find the intercepts.
2
−1
𝑦 = 𝑥2𝑥+3𝑥+2
Asymptotes and Intercepts: Graphing Rational Functions – 8.4
Graphing:


Factor Numerator and denominator. If a factor cancels, it is
a POD, NOT a Vertical Asymptote.
Find all asymptotes and intercepts (see 8.4 page 1)
Example 1: Graph.
y
x
2
x 1
Example 2: Graph.
f ( x) 
x2  6 x  8
x2
Domain is limited to values for which the function is defined.


Set ORIGINAL denominator = 0
The domain is all real numbers EXCEPT the values that
make the denominator 0
PD: ________________
PD: ________________
x-int: _______________
x-int: _______________
y-int: _______________
y-int: _______________
VA: _______________
VA: _______________
HA: ________________
HA: ________________
D: ________________
D: ________________
Graphing and Domain: Graphing Rational Functions – 8.4