<< Chapter < Page Chapter >> Page >
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 ) .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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 )  
Got questions? Get instant answers now!
Got questions? Get instant answers now!

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 )

Got questions? Get instant answers now!

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

if tan alpha + beta is equal to sin x + Y then prove that X square + Y square - 2 I got hyperbole 2 Beta + 1 is equal to zero
Rahul Reply
sin^4+sin^2=1, prove that tan^2-tan^4+1=0
SAYANTANI Reply
what is the formula used for this question? "Jamal wants to save $54,000 for a down payment on a home. How much will he need to invest in an account with 8.2% APR, compounding daily, in order to reach his goal in 5 years?"
Kuz Reply
i don't need help solving it I just need a memory jogger please.
Kuz
A = P(1 + r/n) ^rt
Dale
how to solve an expression when equal to zero
Mintah Reply
its a very simple
Kavita
gave your expression then i solve
Kavita
Hy guys, I have a problem when it comes on solving equations and expressions, can you help me 😭😭
Thuli
Tomorrow its an revision on factorising and Simplifying...
Thuli
ok sent the quiz
kurash
send
Kavita
Hi
Masum
What is the value of log-1
Masum
the value of log1=0
Kavita
Log(-1)
Masum
What is the value of i^i
Masum
log -1 is 1.36
kurash
No
Masum
no I m right
Kavita
No sister.
Masum
no I m right
Kavita
tan20°×tan30°×tan45°×tan50°×tan60°×tan70°
Joju Reply
jaldi batao
Joju
Find the value of x between 0degree and 360 degree which satisfy the equation 3sinx =tanx
musah Reply
what is sine?
tae Reply
what is the standard form of 1
Sanjana Reply
1×10^0
Akugry
Evalute exponential functions
Sujata Reply
30
Shani
The sides of a triangle are three consecutive natural number numbers and it's largest angle is twice the smallest one. determine the sides of a triangle
Jaya Reply
Will be with you shortly
Inkoom
3, 4, 5 principle from geo? sounds like a 90 and 2 45's to me that my answer
Neese
answer is 2, 3, 4
Gaurav
prove that [a+b, b+c, c+a]= 2[a b c]
Ashutosh Reply
can't prove
Akugry
i can prove [a+b+b+c+c+a]=2[a+b+c]
this is simple
Akugry
hi
Stormzy
x exposant 4 + 4 x exposant 3 + 8 exposant 2 + 4 x + 1 = 0
HERVE Reply
x exposent4+4x exposent3+8x exposent2+4x+1=0
HERVE
How can I solve for a domain and a codomains in a given function?
Oliver Reply
ranges
EDWIN
Thank you I mean range sir.
Oliver
proof for set theory
Kwesi Reply
don't you know?
Inkoom
find to nearest one decimal place of centimeter the length of an arc of circle of radius length 12.5cm and subtending of centeral angle 1.6rad
Martina Reply
Practice Key Terms 2

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask