Rational Expressions:
Rational Expression – expression that can be written in the form of P/Q, where P and Q are polynomials and the value of Q is not zero.
Some examples are: -3 y
2
– 1
2 4
5 x + 2
t
2
+ 5t + 6
t + 3
ab
c
2
Just like a rational number, a rational expression represents a division, and so the denominator cannot be 0. A rational expression is said to be undefined for any value of a variable that results in a denominator of zero.
Getting started:
To determine the value of the variable which makes an expression undefined, take the denominator and set it equal to zero, then solve for the variable.
For example: x + 7 This expression would be undefined if x = 6. x - 6
For each of the following state the value of the variable that will make the expression undefined:
1) x – 4
x – 3
2) x + 3
x – 5
3) 5
2x – 8
4) 3
x – 1
8) 3
2x + 6
5) x
x – 2
6) 1
x + 2
7) x
x – 3
9) w
3w + 2
10) 2b
b
2
+ 4b + 3
11) 3j – 9
3j
Multiplication & Division with Rational Expressions:
12) b
b
2
2
– 36
+ 4b – 12
When you multiply fractions, you multiply the numerators and denominators. Than you simplify or reduce if possible. (You can also use the shortcut method!)
4 • 15
5 16
= 4 ∙ 15 = 60 = 3
5 ∙ 16 80 4
The procedure for multiplying rational expressions is similar to the above procedure.
For example:
7x • 3
9 4x 2
= 7x ∙ 3
9 ∙ 4x 2
= 21x =
36x 2
7
12x

***In most cases, using the shortcut method is the easiest. Especially when you are dealing with rational expressions that have polynomial numerators and denominators!
To Multiply or Divide Rational Expressions:
1) If division, take reciprocal of 2 nd
expression.
2) Factor every numerator and denominator (if possible).
3) Apply shortcut by canceling out certain factors.
4) Multiply Across
For example: 3v – 6 • v + 5
5v + 25 9v – 18
= 3(v – 2) • v + 5
5(v + 5) 9(v – 2)
= 1 • 1
5 3
= 1
15
Simplify each expression (Be aware of multiplication or division!):
1) 2 • 1 2) 14x
3
• 6y 3) 7n • 5m
2 n a 5 15y
2
7x
2
45m
3
1
4) 9h
2 
h
3
35 7
5) 6a

3b
5b 5a
6) 3y + 12

y
2
+ 4y
y – 4 2y – 8
7) m – 2

m
2
– 2m – 8 8) 9

6w
3
m
2
+ 6m + 9 m
2
– m – 12 10w
2
9) 5p – 25

p
2
– 2p – 15
p + 3 1
11) u
2
– 3u – 10 • u – 5
u – 2 u 2 – 10u + 25
10) 16r
5s
2
2 
4r
5s
2
12) a
2
– a – 12 • a
2
+ 2a - 3
a 2 – 5a + 4 a 2 + a – 6
13) r
r
2
2
+ 6r + 8 • r
2
+ 3r + 2 r
2
– 4r – 5
– r – 20
14) v
2
– 7v + 10

v
2
– 8v + 15
v
2
+ 2v – 8 v
2
+ 7v + 12
15) x
2
+ x – 6

x
2
– x – 2 16) y
2
+ y – 30

(y + 6)
x 2 – 4x – 21 x 2 – 8x + 7 y 2 + 3y – 10