Section 6.1
Rational Expressions
Definition
A rational expression is the ratio of two polynomials.
Examples:
3x  6
2
x 9
2
4x y
3 2
8x y
bh
2
,
,
8
2
x 1
70
50
Evaluating Rational Expressions
Evaluate 3x  6
x2  9
for
a) x = 0
(a)
(b)
b) x = 3
3(0)  6  6 2


2
(0)  9  9 3
3
3(3)  6 __
=
2
(3)  9
0
Cannot divide by 0
UNDEFINED.
Rational expression is undefined when
its denominator equals to 0
Example
3x  6
Find all numbers for which
x2  9
3x  6
x2  9
is undefined
is undefined when its denominator equal to 0
x 9  0
2
Set denominator equal to 0
( x  3)( x  3)  0
x 3  0
x3
or
or
Factor LHS
x3 0
x  3
Solve for x
Example
4x2
Find all numbers for which 2
is undefined
x  5x  4
4x2
is undefined when its denominator equal to 0
2
x  5x  4
x  5x  4  0
2
( x  1)( x  4)  0
x 1  0
x 1
or
or
Set denominator equal to 0
Factor LHS
x4  0
x4
Solve for x
Simplifying Rational Expressions
Fundamental Property of Rational Expression
AC A

BC B
where A, B, C are polynomials
We can multiply both numerator and denominator by the same polynomial.
We can cancel out any common factors.
A rational expression is in simplified form if its numerator and
Its denominator have no common factors other than 1.
To simplify a rational expression, we
1) Factor the numerator and denominator completely
2) Cancel common factors
Example
4 2
Simplify
4
a3
4 2
20a b
3
25ab
5 1
b
20a b
3
25ab
This expression is already in factored form
1
Just cancel common factors
3
4a

5b
Example
Simplify
6t  24
12
1
6t  24 6(t  4)

12 Factor 12
2
t4

2
Example
x  x6
2
x  3x
2
Simplify
1
x  x  6 ( x  3)( x  2) x  2


2
x( x  3)
x  3x
x
2
1
Factor numerator and denominator completely
Example
Simplify
5  y  1(5  y )

y 5
y 5
1
5 y
y 5
Factor out -1 in the numerator
 1( y  5)  1


= -1
1
y 5
1
Example
Simplify
2( x  3)  2
5( x  3)  5
1
2( x  3)  2
5( x  3)  5
1
No?
NO!
Example
2( x  3)  2
5( x  3)  5
Simplify
2( x  3)  2  2 x  6  2  2 x  4
5( x  3)  5 5 x  15  5 5 x  10
1
Multiply out
2( x  2) 2


5( x  2) 5
1
Example
4 x  4 x  15
3
8 x  50 x
2
Simplify
4 x  4 x  15 (2 x  3)( 2 x  5)

2
3
2 x(4 x  25)
8 x  50 x
2
Factor
1
2x  3
(2 x  3)( 2 x  5)


2 x(2 x  5)( 2 x  5) 2 x(2 x  5)
Difference of 2 squares
1
Example
m  100
2
10m  m
2
a) Evaluate the expression for m = 1
b) Evaluate the expression for m = -10
c) Find all values of m such that the expression
is undefined
d) Simplify the expression
More Examples
1)
24
30
5 x  15
5)
2
x 9
5x6
2)
15 x 2
3y3  6 y2
6)
3y2  9 y4
12a 3b 5 c 2
3)
3ab 7 c 2
3 x 2  14 x  5
7)
3x 2  2 x  1
15  3 y
4) 2
y  5y
ab  3a  5b  15
8)
2
2
15  3a  5b  a b