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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

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
stock therom F=(x2+y2) i-2xy J jaha x=a y=o y=b
Saurabh Reply
sinx sin2x is linearly dependent
cr
root under 3-root under 2 by 5 y square
Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
cosA\1+sinA=secA-tanA
Aasik Reply
Wrong question
Saad
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
what is the best practice that will address the issue on this topic? anyone who can help me. i'm working on my action research.
Melanie Reply
simplify each radical by removing as many factors as possible (a) √75
Jason Reply
how is infinity bidder from undefined?
Karl Reply
what is the value of x in 4x-2+3
Vishal Reply
give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
David Reply
f(x)= 1350. 2^(t/20); where t is in hours.
Merkeb
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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