12.3 : Simplifying Rational Expressions
SPI 3102.3.4 – Operate with, evaluate, and simplify rational expressions including determining
restrictions on the domain of the variables.
Objective: We will be simplifying rational expressions and determining what values of x are excluded.
Rational Expression: an algebraic expression whose numerator and denominator are polynomials.
Excluded Values: values that cause the denominator to be equal to 0.
Ex 1:
Ex 2:
g ≠ -4
Ex 3:
x ≠ __________
You Try:
Ex A:
Ex B:
t≠
y ≠ __________
Ex C:
b ≠ __________
k ≠ __________
To simplify a rational expression:
 You MUST factor both the numerator and the denominator first.
 Then, before you cancel out any like factors you MUST figure out if there are any excluded values of x.
 Then you cancel out any common factors that the denominator and numerator have in common and you are
finished!
Ex 1:
__x + 4__
x2 +3x - 4
___x + 4__
(x + 4)(x – 1)
Step 1: Factor the top and bottom.
The top is already factored so all we have to do is factor
the bottom.
Step 2: Determine what x values are excluded.
(x + 4) = 0 and (x – 1) = 0
-4 -4
+1 +1
x = -4
x = 1
So, x can equal anything except -4 and 1.
___x + 4__
(x + 4)(x – 1)
ANSWER:
__1__
(x – 1)
Step 3: Cancel out any common factors that the numerator and
denominator have.
, where x ≠ -4, 1
Ex 4:
Ans: _________
Set the entire denominator equal to zero and solve. We
do this because any value of x that makes the
denominator zero must be excluded because you can’t
have a zero for the denominator.
Ex 5:
x ≠ _________
Ans: __________ w ≠ _________
Ex 6:
Ans: __________ x ≠ _________
YOU TRY!!!
Ex D:
Ans: _________
Ex E:
r ≠ _________
HW: WB pg. 92 odds (honors: all)
Ans: __________ b ≠ _________
Ex F:
Ans: __________ x ≠ _________
12.4 Multiplying and Dividing Rational Expressions
SPI 3102.3.4- Operate with, evaluate, and simplify rational expressions including determining restrictions on the
domain of the variables.
Objective: Multiply and divide rational expressions.
Multiplication Steps:
 Factor the numerator and the denominator
 Cancel out any common factors that the denominator and numerator have in common
 Multiply the numerators, then multiply the denominators
Example:
(c  4)
45

5
(4c  16)
(c  4)
45

5
4(c  4)
Factor the numerator and the denominator
Cancel out any common factors
9
1 45

5 4
=-
9
4
Ex 1:
Ex 2:
Ex 3:
You Try!!!
Ex A:
Ex B:
Ex C:
Division Steps:
 First, multiply by the reciprocal (flip the second fraction)
 Follow the multiplication steps
Example:
Since it is division you MUST flip the
second fraction and change the
problem to multiplication.
2 g
2 g+4
¸ 2 = ·
g g+4
g
g
2
2 g g
2g
·
=
g g+4
g+4
2
Reduce the numerator and denominator then
multiply across.
Ex 4:
Ex 5:
You Try!!!
Ex D:
Ex E:
HW: WB pg. 93 (#1-6)
EOC PREP:
1) Simplify
2) Simplify the expression below and state all restrictions
on the domain.
x 2  5 x  6 x 2  13x  40

x 2  2 x  15 x 2  6 x  16
for
all values of x for which the expression is defined.
A. 1
B. –1
C.
D.
x 2  11x  24
x 2  11x  24
x 2  11x  24
x 2  11x  24
x2  x  6
x 2  x  12
x4
, x  2, x  4
x2
B. x  3, x  3
x2
, x  4, x  3
C.
x4
x2
,x  0
D.
x4
A.