Rational Expressions: Definition & Simplifying—Explanation
6/05—mm-fd
551
Rational Expressions: Definition & Simplifying—Explanation
Definition
Remember, Rational Number =
Q
P
where P and Q are integers and Q ≠ 0.
Example
12
7
and A polynomial is an expression with finite sum of terms in which all
variables have whole number exponent and no variables appear in
denominator.
Example
42425
2
1
32 yxxyx
We define Rational Expression =
Q
P
where P and Q are Polynomials and Q ≠ 0.
Example
xx
xx
32
54
4
2
where 2x
4
− 3x ≠ 0.
General Rules for Reducing Rational Expressions
Step 1: Factor numerator and denominator as completely as possible.
Step 2: Re-write expression with factors.
Step 3: Cancel any common fractions.
Reducing (Simplifying) a Rational Expression
A rational expression is said to be “reduced to its lowest terms” when the
numerator and denominator have no common factors.
Example 1: Reduce the rational expression
2
( 3)
( 3)( 4 1)
x
x x x
Numerator and denominator have a common factor of (x + 3) which can be
cancelled
)14)(3(
)3(
2
xxx
x
=
14
1
2 xx
Example 2: Simplify the rational expression
2
82
2
x
xx
1. The numerator can be factored to (x + 4) (x - 2)
2. The expression can be re-written as
2
82
2
x
xx
=
2
)2)(4(
x
xx
3. Common factor (x – 2) in numerator and denominator can be
cancelled.
2
82
2
x
xx
=
2
)2)(4(
x
xx
= x + 4
