simplifying rational expressions where p(x) and q(x) are polynomials, q(x) ≠ 0. just as we...
TRANSCRIPT
Simplifying Rational Expressions
P x
Q x
where P(x) and Q(x) are polynomials, Q(x) ≠ 0.
• Just as we simplify rational numbers (fractions) in arithmetic, we want to be able to simplify rational expressions.
• Recall that a rational expression is given by
• To simplify rational expressions:
1. Factor both the numerator and the denominator.
2. Reduce any common factors that are present in both the numerator and the denominator.
• Example 1
Factor the numerator and the denominator.
Simplify the rational expression
2
3 2
4 8
12 24
x x
x x
2
4 ( 2)
12 2
x x
x x
The simplified rational expression is …
Reduce common factors.
2
4 ( 2)
12 2
x x
x x
3 x
2
3 2
x
x x
The variable x found in both the numerator and the denominator cannot be reduced!
Important note:
2
3 2
x
x x
The variable x in the numerator is a term, not a factor. Therefore, it cannot be reduced.
The variable x in the denominator is a term within the factor (x + 2), and is not a factor itself. Therefore, it cannot be reduced.
Remember: we can only reduce common factors.
The only time a term can be reduced is when it is a factor. This only happens when the term makes up the entire numerator or entire denominator.
• Example 2
Since the term 3x is the whole numerator, it is a factor of the numerator, and we can reduce with the 3x factor in the denominator.
Simplify the rational expression
3
3 5
x
x x 1
5x
When we reduce, we are actually removing a factor equal to 1.
The reason behind the reducing is shown by the following:
3
3 5
x
x x
1
5x
3 1
3 5
x
x x
3 1
3 5
x
x x
11
5x
• Example 3
Simplify the rational function, and note any restrictions on the domain.
3
( )3 5
xf x
x x
1( )
5f x
x
To determine the domain restrictions, find the values that will make the original denominator 0:
3
( )3 5
xf x
x x
1( ) , 5, 0
5f x x
x
0x 5x Write your answer as follows:
• Example 4
The 3x in the numerator is just a term now, and not a factor. Therefore we cannot reduce with the 3x in the numerator.
Simplify the rational expression
3 1
3 5
x
x x
Since neither numerator nor denominator can be factored further, and there are no common factors, the expression is simplified.
• Example 5
Simplify the rational expression
3 2
3
2 4 16
3 12
x x x
x x
Factor both numerator and denominator.
2 2 4
3 2 2
x x x
x x x
2
2
2 2 8
3 4
x x x
x x
3 2
3
2 4 16
3 12
x x x
x x
Reduce the common factors.
2 2 4
3 2 2
x x x
x x x
2 4
3 2
x
x
• Example 6
Simplify the rational expression
4
4
x
x
Since the x’s and 4’s are terms, they can’t be reduced.
At first glance, one would think that the expression is simplified.
Factor a negative one out of the denominator.
4
4
x
x
Reduce the common binomial factors to get …
4
( 4 )
x
x
4
( 4)
x
x
1
1
1
