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Here are a few recommended readings before getting started with this lesson.
A rational expression is a fraction where both the numerator and the denominator are polynomials.
q(x)p(x)
Rational Expressions | |
---|---|
Not in Simplest Form | In Simplest Form |
x(x−3)(y+2)xy | x+1x−1 |
x2+1x4+x2 | x2−x−6x3+7 |
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. Any 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 |
x2−x−6x3+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 |
x2+1x4+x2 | There is no real number that makes x2+1 zero | None |
Simplifying a rational expression can remove some of the excluded values that appear in the original expression. A rational expression and its simplified form must have the same domain in order for them to be equivalent expressions. This means that the excluded values that are no longer visible in the simplified expression must still be declared.
Equivalent Expressions | |
---|---|
Rational Expression | Simplified Form |
(x+2)(x−3)x−3,x=-2,3 | x+21,x=-2,3 |
x2−1x2+2x+1,x=-1,1 | x−1x+1,x=-1,1 |
x2x3−2x2+x,x=0 | xx2−2x+1,x=0 |
A rational expression is undefined when its denominator is 0. The values that make the denominator of a rational expression equal to 0 are called excluded values because they are excluded from its domain. Determine the excluded values for the indicated rational expressions.
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)
A rational expression is undefined for values that make its denominator zero. Therefore, those values should be excluded from the domain.
Use the Zero Product Property
(I): LHS−7=RHS−7
(II): LHS+5=RHS+5
Operations with rational numbers and rational expressions are similar.
Multiplying rational expressions works the same way as multiplying fractions. The numerators and denominators are multiplied separately.
Q(x)P(x)⋅G(x)H(x)=Q(x)⋅G(x)P(x)⋅H(x)
Multiply fractions
Cancel out common factors
Simplify quotient
a⋅a=a2
Dividing two rational expressions is the same as multiplying the first expression by the reciprocal of the second expression.
Factor out -1
Cancel out common factors
Simplify quotient
Distribute -1
Ramsha drew the plan of her house and labeled the sides, measured in meters, as shown.
ℓ=x2+18x+812x2−6x, w=x2−99x+81
Factor out 2x
Split into factors
Commutative Property of Multiplication
Write as a power
a2+2ab+b2=(a+b)2
Factor out 9