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In this section students will:
  • Simplify rational expressions.
  • Multiply rational expressions.
  • Divide rational expressions.
  • Add and subtract rational expressions.
  • Simplify complex rational expressions.

A pastry shop has fixed costs of $ 280 per week and variable costs of $ 9 per box of pastries. The shop’s costs per week in terms of x , the number of boxes made, is 280 + 9 x . We can divide the costs per week by the number of boxes made to determine the cost per box of pastries.

280 + 9 x x

Notice that the result is a polynomial expression divided by a second polynomial expression. In this section, we will explore quotients of polynomial expressions.

Simplifying rational expressions

The quotient of two polynomial expressions is called a rational expression    . We can apply the properties of fractions to rational expressions, such as simplifying the expressions by canceling common factors from the numerator and the denominator. To do this, we first need to factor both the numerator and denominator. Let’s start with the rational expression shown.

x 2 + 8 x + 16 x 2 + 11 x + 28

We can factor the numerator and denominator to rewrite the expression.

( x + 4 ) 2 ( x + 4 ) ( x + 7 )

Then we can simplify that expression by canceling the common factor ( x + 4 ) .

x + 4 x + 7

Given a rational expression, simplify it.

  1. Factor the numerator and denominator.
  2. Cancel any common factors.

Simplifying rational expressions

Simplify x 2 9 x 2 + 4 x + 3 .

( x + 3 ) ( x 3 ) ( x + 3 ) ( x + 1 ) Factor the numerator and the denominator . x 3 x + 1 Cancel common factor  ( x + 3 ) .
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Can the x 2 term be cancelled in [link] ?

No. A factor is an expression that is multiplied by another expression. The x 2 term is not a factor of the numerator or the denominator.

Simplify x 6 x 2 36 .

1 x + 6

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Multiplying rational expressions

Multiplication of rational expressions works the same way as multiplication of any other fractions. We multiply the numerators to find the numerator of the product, and then multiply the denominators to find the denominator of the product. Before multiplying, it is helpful to factor the numerators and denominators just as we did when simplifying rational expressions. We are often able to simplify the product of rational expressions.

Given two rational expressions, multiply them.

  1. Factor the numerator and denominator.
  2. Multiply the numerators.
  3. Multiply the denominators.
  4. Simplify.

Multiplying rational expressions

Multiply the rational expressions and show the product in simplest form:

( x + 5 ) ( x 1 ) 3 ( x + 6 ) ( 2 x 1 ) ( x + 5 )
( x + 5 ) ( x 1 ) 3 ( x + 6 ) ( 2 x 1 ) ( x + 5 ) Factor the numerator and denominator . ( x + 5 ) ( x 1 ) ( 2 x 1 ) 3 ( x + 6 ) ( x + 5 ) Multiply numerators and denominators . ( x + 5 ) ( x 1 ) ( 2 x 1 ) 3 ( x + 6 ) ( x + 5 ) Cancel common factors to simplify . ( x 1 ) ( 2 x 1 ) 3 ( x + 6 )  
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Multiply the rational expressions and show the product in simplest form:

x 2 + 11 x + 30 x 2 + 5 x + 6 x 2 + 7 x + 12 x 2 + 8 x + 16

( x + 5 ) ( x + 6 ) ( x + 2 ) ( x + 4 )

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Dividing rational expressions

Division of rational expressions works the same way as division of other fractions. To divide a rational expression by another rational expression, multiply the first expression by the reciprocal of the second. Using this approach, we would rewrite 1 x ÷ x 2 3 as the product 1 x 3 x 2 . Once the division expression has been rewritten as a multiplication expression, we can multiply as we did before.

Questions & Answers

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Mark Reply
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Miranda
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Miranda Drice
jai
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jai
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Miranda
I am living in india
jai
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Miranda
what is the formula for calculating algebraic
Propessor Reply
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
sita Reply
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
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jai
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Propessor
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What is algebra
Pearl Reply
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math. I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
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Jeffrey
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Miranda
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Miranda
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Jeffrey
how about you? what grade are you now?
Jeffrey
I'm going to 11grade
Miranda
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Miranda
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Steve
if you never use it you won't be able to appreciate it
Steve
I don't know why. But Im trying to like it.
Jeffrey
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Jeffrey
so you better
Miranda
what is the solution of the given equation?
Nelson Reply
which equation
Miranda
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Jeffrey
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Miranda
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Jeffrey
answer and questions in exercise 11.2 sums
Yp Reply
how do u calculate inequality of irrational number?
Alaba
give me an example
Chris
and I will walk you through it
Chris
cos (-z)= cos z .
Swadesh
what is a algebra
Jallah Reply
(x+x)3=?
Narad
6x
Obed
what is the identity of 1-cos²5x equal to?
liyemaikhaya Reply
__john __05
Kishu
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Nokwanda
C'est comment
Abdel
Hi
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SORIE
Hiiii
Chinni
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ANSHU
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SORIE
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Abdel
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Yaona
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SORIE
it's 12
what is the function of sine with respect of cosine , graphically
Karl Reply
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
Aashish Reply
sinx sin2x is linearly dependent
cr Reply
what is a reciprocal
Ajibola Reply
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
 Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
Funmilola Reply
I don't understand how radicals works pls
Kenny Reply
How look for the general solution of a trig function
collins Reply
Practice Key Terms 2

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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