# 1.6 Rational expressions

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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 $\text{\hspace{0.17em}}\text{}280\text{\hspace{0.17em}}$ per week and variable costs of $\text{\hspace{0.17em}}\text{}9\text{\hspace{0.17em}}$ per box of pastries. The shop’s costs per week in terms of $\text{\hspace{0.17em}}x,$ the number of boxes made, is $\text{\hspace{0.17em}}280+9x.\text{\hspace{0.17em}}$ We can divide the costs per week by the number of boxes made to determine the cost per box of pastries.

$\frac{280+9x}{x}$

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.

$\frac{{x}^{2}+8x+16}{{x}^{2}+11x+28}$

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

$\frac{{\left(x+4\right)}^{2}}{\left(x+4\right)\left(x+7\right)}$

Then we can simplify that expression by canceling the common factor $\text{\hspace{0.17em}}\left(x+4\right).$

$\frac{x+4}{x+7}$

Given a rational expression, simplify it.

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

## Simplifying rational expressions

Simplify $\text{\hspace{0.17em}}\frac{{x}^{2}-9}{{x}^{2}+4x+3}.$

Can the $\text{\hspace{0.17em}}{x}^{2}\text{\hspace{0.17em}}$ term be cancelled in [link] ?

No. A factor is an expression that is multiplied by another expression. The $\text{\hspace{0.17em}}{x}^{2}\text{\hspace{0.17em}}$ term is not a factor of the numerator or the denominator.

Simplify $\text{\hspace{0.17em}}\frac{x-6}{{x}^{2}-36}.$

$\frac{1}{x+6}$

## 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:

$\frac{\left(x+5\right)\left(x-1\right)}{3\left(x+6\right)}\cdot \frac{\left(2x-1\right)}{\left(x+5\right)}$

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

$\frac{{x}^{2}+11x+30}{{x}^{2}+5x+6}\cdot \frac{{x}^{2}+7x+12}{{x}^{2}+8x+16}$

$\frac{\left(x+5\right)\left(x+6\right)}{\left(x+2\right)\left(x+4\right)}$

## 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 $\text{\hspace{0.17em}}\frac{1}{x}÷\frac{{x}^{2}}{3}\text{\hspace{0.17em}}$ as the product $\text{\hspace{0.17em}}\frac{1}{x}\cdot \frac{3}{{x}^{2}}.\text{\hspace{0.17em}}$ Once the division expression has been rewritten as a multiplication expression, we can multiply as we did before.

#### Questions & Answers

find general solution of the Tanx=-1/root3,secx=2/root3
Nani Reply
find general solution of the following equation
Nani
the value of 2 sin square 60 Cos 60
Sanjay Reply
0.75
Lynne
0.75
Inkoom
when can I use sin, cos tan in a giving question
duru Reply
depending on the question
Nicholas
I am a carpenter and I have to cut and assemble a conventional roof line for a new home. The dimensions are: width 30'6" length 40'6". I want a 6 and 12 pitch. The roof is a full hip construction. Give me the L,W and height of rafters for the hip, hip jacks also the length of common jacks.
John
I want to learn the calculations
Koru Reply
where can I get indices
Kojo Reply
I need matrices
Nasasira
hi
Raihany
Hi
Solomon
need help
Raihany
maybe provide us videos
Nasasira
about complex fraction
Raihany
Hello
Cromwell
a
Amie
What do you mean by a
Cromwell
nothing. I accidentally press it
Amie
you guys know any app with matrices?
Khay
Ok
Cromwell
Solve the x? x=18+(24-3)=72
Leizel Reply
x-39=72 x=111
Suraj
Solve the formula for the indicated variable P=b+4a+2c, for b
Deadra Reply
Need help with this question please
Deadra
b=-4ac-2c+P
Denisse
b=p-4a-2c
Suddhen
b= p - 4a - 2c
Snr
p=2(2a+C)+b
Suraj
b=p-2(2a+c)
Tapiwa
P=4a+b+2C
COLEMAN
b=P-4a-2c
COLEMAN
like Deadra, show me the step by step order of operation to alive for b
John
A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
Kaitlyn Reply
The sequence is {1,-1,1-1.....} has
amit Reply
circular region of radious
Kainat Reply
how can we solve this problem
Joel Reply
Sin(A+B) = sinBcosA+cosBsinA
Eseka Reply
Prove it
Eseka
Please prove it
Eseka
hi
Joel
yah
immy
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
Arleathia Reply
7.5 and 37.5
Nando
how would this look as an equation?
Hayden
5x+x=45
Khay
find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
Vedant
the 28th term is 175
Nando
192
Kenneth
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
SANDESH Reply

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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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