In Simplest Form | Not in Simplest Form |
---|---|
$x+1x−1 $ | $x(x−3)(y+2)xy $ |
$x_{2}−x−6x_{3}+7 $ | $x_{2}+1x_{4}+x_{2} $ |
Notice that for some of the expressions shown in the table, there are some x-values that make the denominator 0. For example, the denominator of $x+1x−1 $ is 0 when x=-1. A value of a variable for which a rational expression is undefined is called an excluded value.
Expression | Restriction | Excluded Value(s) |
---|---|---|
$x+1x−1 $ | x+1≠0 | x≠-1 |
$x_{2}−x−6x_{3}+7 $ | x2−x−6≠0 | x≠-2 and x≠3 |
$x(x−3)(y+2)xy $ | x(x−3)(y+2)≠0 | x≠0, x≠3, and y≠-2 |
$x_{2}+1x_{4}+x_{2} $ | There is no x-value that makes x2+1 zero | None |
A rational expression is in simplest form if the numerator and denominator have no factors in common. Rational expressions that are not in simplest form can be simplified by canceling out the common factors in the numerator and the denominator. Common factors can be found through various ways of factoring polynomials.
Split into factors
Factor out x
Write as a power
a2−b2=(a+b)(a−b)
Split into factors
Write as a power
a2−2ab+b2=(a−b)2
Commutative Property of Multiplication
a−b=-(b−a)
Since rational expressions are essentially fractions, it's possible to add, subtract, multiply and divide them. When creating a multiple of a rational expression, $q(x)p(x) ,$ by multiplying by the factor k in both the denominator and the numerator, the equality still holds true.
Canceling out a factor k, yields the same equality. In both cases, the factor k can take all values except 0.
Operations with rational numbers and rational expressions are similar.
Factor out 7
Multiply fractions
Cancel out common factors
Simplify quotient
Multiply
Factor out -1
Cancel out common factors
Simplify quotient
Multiply
Write as a power
a2−b2=(a+b)(a−b)
$ba =b/(x+1)a/(x+1) $
$ba /dc =ba ⋅cd $
Multiply fractions
When adding and subtracting rational expressions, the same rules apply as when adding and subtracting fractions. If they share a denominator, the numerators can be added or subtracted directly.
$ba =b⋅(x−1)a⋅(x−1) $
(a+b)(a−b)=a2−b2
Calculate power
Distribute x
$ba =b⋅xa⋅x $
Multiply
Distribute x
Add fractions
Simplify terms