Manipulating Rational Expressions

A rational number is a number that can be written as a fraction of two integers. Similarly, a rational expression is an expression written as a fraction of two polynomials. Because of the similarity with rational numbers, many of the same rules apply.

Concept

Rational Expression

A rational expression is a fraction where both the numerator and the denominator are polynomials. The expression below is an example of a rational expression.

A rational expression is in simplest form when its numerator and denominator have no common factors.

In Simplest Form	Not in Simplest Form
$\frac{x - 1}{x + 1}$	$\frac{x y}{x ( x - 3 ) ( y + 2 )}$
$\frac{x ^{3} + 7}{x ^{2} - x - 6}$	$\frac{x ^{4} + x ^{2}}{x ^{2} + 1}$

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 $\frac{x - 1}{x + 1}$ 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)
$\frac{x - 1}{x + 1}$	x+1≠0	x≠-1
$\frac{x ^{3} + 7}{x ^{2} - x - 6}$	x2−x−6≠0	x≠-2 and x≠3
$\frac{x y}{x ( x - 3 ) ( y + 2 )}$	x(x−3)(y+2)≠0	x≠0, x≠3, and y≠-2
$\frac{x ^{4} + x ^{2}}{x ^{2} + 1}$	There is no x-value that makes x2+1 zero	None

Method

Simplifying a Rational Expression

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.

The rational expression can be simplified in three steps.

1

Factor the Numerator and Denominator

Check first if either the numerator or denominator (or both) can be factored by GCF. In this example, the factor x is in both terms in the denominator.

\frac{x ^{2} - 6 x + 9}{9 x - x ^{3}}

SplitIntoFactors

Split into factors

\frac{x ^{2} - 6 x + 9}{x \cdot 9 - x \cdot x ^{2}}

FactorOut

Factor out x

\frac{x ^{2} - 6 x + 9}{x ( 9 - x ^{2} )}

Determine if it is possible to factor either the numerator or the denominator using the difference of squares. Look for an expression in the form a2−b2, which can be factored as (a+b)(a−b). In the example, (9−x)2 can be factored using this rule.

\frac{x ^{2} - 6 x + 9}{x ( 9 - x ^{2} )}

WritePow

Write as a power

\frac{x ^{2} - 6 x + 9}{x ( 3 ^{2} - x ^{2} )}

FacDiffSquares

a2−b2=(a+b)(a−b)

\frac{x ^{2} - 6 x + 9}{x ( 3 + x ) ( 3 - x )}

See if either the numerator or denominator is a perfect square trinomial. Look for an expression in the form a2±2ab+b2, which can be factored as (a±b)2. Here, the numerator is a perfect square trinomial.

\frac{x ^{2} - 6 x + 9}{x ( 3 + x ) ( 3 - x )}

SplitIntoFactors

Split into factors

\frac{x ^{2} - 2 \cdot x \cdot 3 + 9}{x ( 3 + x ) ( 3 - x )}

WritePow

Write as a power

\frac{x ^{2} - 2 \cdot x \cdot 3 + 3 ^{2}}{x ( 3 + x ) ( 3 - x )}

FacNegPerfectSquare

a2−2ab+b2=(a−b)2

\frac{( x - 3 ) ^{2}}{x ( 3 + x ) ( 3 - x )}

In some cases, a negative sign can be factored out for one expression to have the same form as another factor. Here, the factor 3−x in the denominator is almost identical with the factor x−3 in the numerator.

\frac{( x - 3 ) ^{2}}{x ( 3 + x ) ( 3 - x )}

CommutativePropMult

Commutative Property of Multiplication

\frac{( x - 3 ) ^{2}}{( 3 - x ) \cdot x \cdot ( 3 + x )}

FactorOutNegSignSwap

a−b=-(b−a)

\frac{( x - 3 ) ^{2}}{- ( x - 3 ) \cdot x \cdot ( 3 + x )}

2

List Restricted Values

Before simplifying the common factors, check if there are any restrictions on x.

Notice that the expression is undefined when x=-3, 0, or 3. Additionally, (x−3) is the common factor to the numerator and denominator.

3

Cancel Out Common Factors

Finally, the common factor is eliminated.

\frac{( x - 3 ) ^{2}}{- ( x - 3 ) \cdot x \cdot ( 3 + x )}

ReduceFrac

$\frac{a}{b} = \frac{a / ( x - 3 )}{b / ( x - 3 )}$

\frac{x - 3}{- x ( 3 + x )}

The rational expression is now written in its simplest form. To make the given expression and the simplified expression equivalent, the domain of the simplified expression should be restricted by excluding x=3.

Since the restriction x≠3 cannot be seen from the simplified expression, it is noted. The other restrictions are also evident from the simplified expression.

Simplify the rational expression

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Write the rational expression in its simplest form.

Show Solution

To begin, we'll factor the numerator completely. Notice that the denominator is already written in factored form. The numerator contains the expression 3x2−75. Since 3 is a factor of 75, we can factor 3 out of the entire expression.

3x2−75

FactorOut

Factor out 3

3 (x^{2} - 25)

Notice that x2−25 is a difference of squares, which means it can be factored to be (x+5)(x−5). Thus, the given expression can be written as

We can see that the common factors between the numerator and denominator are 3 and (x+5). Thus, we can cancel them out. Writing the expression in its simplest form gives

Rule

Manipulating Rational Expressions

Since rational expressions are essentially fractions, it's possible to add, subtract, multiply and divide them. When creating a multiple of a rational expression, $\frac{p ( x )}{q ( x )},$ 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.

It is also possible to create a multiple or cancel out a factor by using a more complex polynomial:

Consider the domain. The first expression is undefined for x=1, but the second expression is not. It looks like the domain has been expanded when (x−1) was canceled out, but this is not the case. For the equality to hold true, all x-values must give the same value on both sides. Taking this into account gives

Rule

Multiplying and Dividing Rational Expressions

Operations with rational numbers and rational expressions are similar.

Method

Multiplying Rational Expressions

Multiplying rational expressions works the same way as multiplying fractions. The numerators and denominators are multiplied separately.

Consider the following product.

To multiply the rational expressions, these three steps can be followed.

1

Factor Each Numerator and Denominator

Before multiplying, it is helpful to factor the numerators and denominators, if possible. Start with the first rational expression.

\frac{x ^{2} - 2 x - 1 5}{7 x + 4 2}

Factor the numerator

WriteSum

Write as a sum

\frac{x ^{2} - 5 x + 3 x - 1 5}{7 x + 4 2}

FactorOut

Factor out x

\frac{x ( x - 5 ) + 3 x - 1 5}{7 x + 4 2}

FactorOut

Factor out 3

\frac{x ( x - 5 ) + 3 ( x - 5 )}{7 x + 4 2}

FactorOut

Factor out (x−5)

\frac{( x + 3 ) ( x - 5 )}{7 x + 4 2}

Factor the denominator

FactorOut

Factor out 7

\frac{( x + 3 ) ( x - 5 )}{7 ( x + 6 )}

Now, the numerator and denominator of the second expression will be factored.

\frac{x ^{2} + 9 x + 1 8}{x ^{2} - 1 1 + 3 0}

Factor the numerator

WriteSum

Write as a sum

\frac{x ^{2} + 6 x + 3 x + 1 8}{x ^{2} - 1 1 + 3 0}

FactorOut

Factor out x

\frac{x ( x + 6 ) + 3 x + 1 8}{x ^{2} - 1 1 x + 3 0}

FactorOut

Factor out 3

\frac{x ( x + 6 ) + 3 ( x + 6 )}{x ^{2} - 1 1 x + 3 0}

FactorOut

Factor out (x+6)

\frac{( x + 3 ) ( x + 6 )}{x ^{2} - 1 1 x + 3 0}

Factor the denominator

WriteDiff

Write as a difference

\frac{( x + 3 ) ( x + 6 )}{x ^{2} - 6 x - 5 x + 3 0}

FactorOut

Factor out x

\frac{( x + 3 ) ( x + 6 )}{x ( x - 6 ) - 5 x + 3 0}

FactorOut

Factor out -5

\frac{( x + 3 ) ( x + 6 )}{x ( x - 6 ) - 5 ( x - 6 )}

FactorOut

Factor out (x−6)

\frac{( x + 3 ) ( x + 6 )}{( x - 5 ) ( x - 6 )}

2

Multiply the Numerators and Multiply the Denominators

Now that both expressions have been factored, the numerators and denominators can be multiplied.

\frac{( x + 3 ) ( x - 5 )}{7 ( x + 6 )} \cdot \frac{( x + 3 ) ( x + 6 )}{( x - 5 ) ( x - 6 )}

MultFrac

Multiply fractions

\frac{( x + 3 ) ( x - 5 ) ( x + 3 ) ( x + 6 )}{7 ( x + 6 ) ( x - 5 ) ( x - 6 )}

Note that the product is undefined when x=-6, 5, or 6. Furthermore,

(x - 5)

and (x+6) are the common factors of the numerator and denominator.

4

Cancel Out Common Factors

Finally, the product is simplified by dividing out the common factors.

\frac{( x + 3 ) ( x - 5 ) ( x + 3 ) ( x + 6 )}{7 ( x + 6 ) ( x - 5 ) ( x - 6 )}

CancelCommonFac

Cancel out common factors

\frac{( x + 3 ) ( x - 5 ) ( x + 3 ) ( x + 6 )}{7 ( x + 6 ) ( x - 5 ) ( x - 6 )}

SimpQuot

Simplify quotient

\frac{( x + 3 ) ( x + 3 )}{7 ( x - 6 )}

Multiply

\frac{( x + 3 ) ^{2}}{7 ( x - 6 )}

Considering the denominator of the simplified expression and any other denominator used, the values -6, 5, and 6 must be excluded from the domain of the simplified expression.

Method

Dividing Rational Expressions

To divide two rational expressions, the first step is to multiply the first expression by the reciprocal of the second expression, and then multiply, similar to dividing fractions.

As an example, the quotient will be calculated.

The process of dividing rational expression can be completed in four steps.

1

Rewrite the Division as the Product of the Dividend and the Reciprocal of the Divisor

To divide a rational expression by another rational expression, multiply the first expression by the reciprocal of the second.

Once the division expression is rewritten as a multiplication expression, the remaining steps are the same as those for multiplying rational expressions.

2

Factor Each Numerator and Denominator

Factor the numerators and denominators, if possible. In this case, only the denominator of the first rational expression can be factored.

\frac{5 - x}{x ^{2} + 3 x}

FactorOut

Factor out x

\frac{5 - x}{x ( x + 3 )}

Now, the numerator and denominator of the other expression will be factored.

\frac{x ^{2} + 1 0 x + 2 1}{x ^{2} - 2 5}

Factor the numerator

WriteSum

Write as a sum

\frac{x ^{2} + 3 x + 7 x + 2 1}{x ^{2} - 2 5}

FactorOut

Factor out x

\frac{x ( x + 3 ) + 7 x + 2 1}{x ^{2} - 2 5}

FactorOut

Factor out 3

\frac{x ( x + 3 ) + 7 ( x + 3 )}{x ^{2} - 2 5}

FactorOut

Factor out (x+3)

\frac{( x + 3 ) ( x + 7 )}{x ^{2} - 2 5}

Factor the denominator

WritePow

Write as a power

\frac{( x + 3 ) ( x + 7 )}{x ^{2} - 5 ^{2}}

ExpandDiffSquares

(a+b)(a−b)=a2−b2

\frac{( x + 3 ) ( x + 7 )}{( x + 5 ) ( x - 5 )}

3

Multiply the Numerators and Multiply the Denominators

Now the numerators and denominators can be multiplied.

Note that this product is undefined when x=-5, -3, 0 or 5. At this point, the values that make the denominator of the divisor, which are also the values that make the numerator of the second expression in the product, should be considered. Those values are -7 and -3.

Excluded Values - 7, - 5, - 3, 0, 5

4

Cancel Out Common Factors

Finally, the product is simplified by dividing out common factors.

\frac{( 5 - x ) ( x + 3 ) ( x + 7 )}{x ( x + 3 ) ( x + 5 ) ( x - 5 )}

FactorOut

Factor out -1

\frac{- 1 ( x - 5 ) ( x + 3 ) ( x + 7 )}{x ( x + 3 ) ( x + 5 ) ( x - 5 )}

CancelCommonFac

Cancel out common factors

\frac{- 1 ( x - 5 ) ( x + 3 ) ( x + 7 )}{x ( x + 3 ) ( x + 5 ) ( x - 5 )}

SimpQuot

Simplify quotient

\frac{- 1 ( x + 7 )}{x ( x + 5 )}

Multiply

\frac{- x - 7}{x ( x + 5 )}

Considering the denominator of the simplified expression and any other denominator used, the values -7, -5, -3, 0, and 5 must be excluded from the domain of the expression.

Multiply and divide the rational expressions

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Determine the product and quotient of the rational expressions. Simplify completely.

\frac{x ^{2} - 1}{x + 2} and \frac{2 x}{x + 1}

Show Solution

Let's start with multiplying the expressions. The numerators and denominators are multiplied separately.

\frac{x ^{2} - 1}{x + 2} \cdot \frac{2 x}{x + 1}

MultFrac

Multiply fractions

\frac{( x ^{2} - 1 ) \cdot 2 x}{( x + 2 ) \cdot ( x + 1 )}

Since we want to simplify the expression, let's factor the numerator. (x2−1) is a difference of squares. We can factor it, and then cancel out any common factors between the numerator and the denominator.

\frac{( x ^{2} - 1 ) \cdot 2 x}{( x + 2 ) \cdot ( x + 1 )}

WritePow

Write as a power

\frac{( x ^{2} - 1 ^{2} ) \cdot 2 x}{( x + 2 ) \cdot ( x + 1 )}

FacDiffSquares

a2−b2=(a+b)(a−b)

\frac{( x + 1 ) ( x - 1 ) \cdot 2 x}{( x + 2 ) \cdot ( x + 1 )}

ReduceFrac

$\frac{a}{b} = \frac{a / ( x + 1 )}{b / ( x + 1 )}$

\frac{( x - 1 ) \cdot 2 x}{x + 2}

The product is

\frac{2 x ( x - 1 )}{x + 2} .

However, since we divided out the common factor x+1 we have to take that into account:

Now, we divide the expressions.

\frac{x ^{2} - 1}{x + 2} / \frac{2 x}{x + 1}

DivFracByFracSL

$\frac{a}{b} / \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$

\frac{x ^{2} - 1}{x + 2} \cdot \frac{x + 1}{2 x}

MultFrac

Multiply fractions

\frac{( x ^{2} - 1 ) ( x + 1 )}{( x + 2 ) \cdot 2 x}

From the multiplication, we know that x2−1 can be written as the factors x+1 and x−1. Therefore, the numerator and denominator do not have any common factors. The simplest form of the quotient is

Rule

Adding and Subtracting Rational Expressions

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.

However, if the denominators are different, the expressions have to be manipulated to find a common denominator. One way of doing that is to multiply both numerator and denominator of one of the rational expressions with the denominator of the other, and vice versa.

Add the rational expressions

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Determine the sum of the rational expressions.

\frac{x + 1}{x} and \frac{2 x}{x - 1}

Show Solution

Similar to fractions, in order to add rational expressions they must have common denominators. Here, the denominators are not the same. This means we need to manipulate the expressions before adding them. We multiply the numerator and denominator of each expression by the denominator in the other expression.