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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

bsc F. y algebra and trigonometry pepper 2
Aditi Reply
given that x= 3/5 find sin 3x
Adamu Reply
4
DB
remove any signs and collect terms of -2(8a-3b-c)
Joeval Reply
-16a+6b+2c
Will
is that a real answer
Joeval
(x2-2x+8)-4(x2-3x+5)
Ayush Reply
sorry
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
(X2-2X+8)-4(X2-3X+5)=0 ?
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
Y
master
X2-2X+8-4X2+12X-20=0 (X2-4X2)+(-2X+12X)+(-20+8)= 0 -3X2+10X-12=0 3X2-10X+12=0 Use quadratic formula To find the answer answer (5±Root11i)/3
master
Soo sorry (5±Root11* i)/3
master
x2-2x+8-4x2+12x-20 x2-4x2-2x+12x+8-20 -3x2+10x-12 now you can find the answer using quadratic
Mukhtar
explain and give four example of hyperbolic function
Lukman Reply
What is the correct rational algebraic expression of the given "a fraction whose denominator is 10 more than the numerator y?
Racelle Reply
y/y+10
Mr
Find nth derivative of eax sin (bx + c).
Anurag Reply
Find area common to the parabola y2 = 4ax and x2 = 4ay.
Anurag
A rectangular garden is 25ft wide. if its area is 1125ft, what is the length of the garden
Jhovie Reply
to find the length I divide the area by the wide wich means 1125ft/25ft=45
Miranda
thanks
Jhovie
What do you call a relation where each element in the domain is related to only one value in the range by some rules?
Charmaine Reply
A banana.
Yaona
given 4cot thither +3=0and 0°<thither <180° use a sketch to determine the value of the following a)cos thither
Snalo Reply
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Mark Reply
nothing up todat yet
Miranda
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jai
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jai
Miranda Drice
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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
hi
jai
hi Miranda
jai
thanks
Propessor
welcome
jai
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
lolll who told you I'm good at it
Miranda
something seems to wispher me to my ear that u are good at it. lol
Jeffrey
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Miranda
but seriously, Im really bad at math. And I hate it. But you see, I downloaded this app two months ago hoping to master it.
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
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Steve
I don't know why. But Im trying to like it.
Jeffrey
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Jeffrey
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Miranda
what is the solution of the given equation?
Nelson Reply
which equation
Miranda
I dont know. lol
Jeffrey
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Miranda
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Jeffrey
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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