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1.6 Rational expressions  (Page 2/6)

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1 x ⋅ 3 x 2 = 3 x 3

Given two rational expressions, divide them.

  1. Rewrite as the first rational expression multiplied by the reciprocal of the second.
  2. Factor the numerators and denominators.
  3. Multiply the numerators.
  4. Multiply the denominators.
  5. Simplify.

Dividing rational expressions

Divide the rational expressions and express the quotient in simplest form:

2 x 2 + x − 6 x 2 − 1 ÷ x 2 − 4 x 2 + 2 x + 1
9 x 2 − 16 3 x 2 + 17 x − 28 ÷ 3 x 2 − 2 x − 8 x 2 + 5 x − 14
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Divide the rational expressions and express the quotient in simplest form:

9 x 2 − 16 3 x 2 + 17 x − 28 ÷ 3 x 2 − 2 x − 8 x 2 + 5 x − 14

1

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Adding and subtracting rational expressions

Adding and subtracting rational expressions works just like adding and subtracting numerical fractions. To add fractions, we need to find a common denominator. Let’s look at an example of fraction addition.

5 24 + 1 40 = 25 120 + 3 120 = 28 120 = 7 30

We have to rewrite the fractions so they share a common denominator before we are able to add. We must do the same thing when adding or subtracting rational expressions.

The easiest common denominator to use will be the least common denominator    , or LCD. The LCD is the smallest multiple that the denominators have in common. To find the LCD of two rational expressions, we factor the expressions and multiply all of the distinct factors. For instance, if the factored denominators were   ( x + 3 ) ( x + 4 )   and   ( x + 4 ) ( x + 5 ) , then the LCD would be   ( x + 3 ) ( x + 4 ) ( x + 5 ) .

Once we find the LCD, we need to multiply each expression by the form of 1 that will change the denominator to the LCD. We would need to multiply the expression with a denominator of   ( x + 3 ) ( x + 4 )   by   x + 5 x + 5   and the expression with a denominator of   ( x + 4 ) ( x + 5 )   by   x + 3 x + 3 .

Given two rational expressions, add or subtract them.

  1. Factor the numerator and denominator.
  2. Find the LCD of the expressions.
  3. Multiply the expressions by a form of 1 that changes the denominators to the LCD.
  4. Add or subtract the numerators.
  5. Simplify.

Adding rational expressions

Add the rational expressions:

5 x + 6 y

First, we have to find the LCD. In this case, the LCD will be   x y .   We then multiply each expression by the appropriate form of 1 to obtain   x y   as the denominator for each fraction.

5 x ⋅ y y + 6 y ⋅ x x 5 y x y + 6 x x y

Now that the expressions have the same denominator, we simply add the numerators to find the sum.

6 x + 5 y x y
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Subtracting rational expressions

Subtract the rational expressions:

6 x 2 + 4 x + 4 − 2 x 2 −4
6 ( x + 2 ) 2 − 2 ( x + 2 ) ( x − 2 ) Factor . 6 ( x + 2 ) 2 ⋅ x − 2 x − 2 − 2 ( x + 2 ) ( x − 2 ) ⋅ x + 2 x + 2 Multiply each fraction to get LCD as denominator . 6 ( x − 2 ) ( x + 2 ) 2 ( x − 2 ) − 2 ( x + 2 ) ( x + 2 ) 2 ( x − 2 ) Multiply . 6 x − 12 − ( 2 x + 4 ) ( x + 2 ) 2 ( x − 2 ) Apply distributive property . 4 x − 16 ( x + 2 ) 2 ( x − 2 ) Subtract . 4 ( x − 4 ) ( x + 2 ) 2 ( x − 2 ) Simplify .
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Do we have to use the LCD to add or subtract rational expressions?

No. Any common denominator will work, but it is easiest to use the LCD.

Subtract the rational expressions:   3 x + 5 − 1 x −3 .

2 ( x −7 ) ( x + 5 ) ( x −3 )

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Simplifying complex rational expressions

A complex rational expression is a rational expression that contains additional rational expressions in the numerator, the denominator, or both. We can simplify complex rational expressions by rewriting the numerator and denominator as single rational expressions and dividing. The complex rational expression   a 1 b + c   can be simplified by rewriting the numerator as the fraction   a 1   and combining the expressions in the denominator as   1 + b c b .   We can then rewrite the expression as a multiplication problem using the reciprocal of the denominator. We get   a 1 ⋅ b 1 + b c , which is equal to   a b 1 + b c .

Questions & Answers

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
Kate Reply
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Kate Reply
what is the change in momentum of a body?
Eunice Reply
what is a capacitor?
Raymond Reply
Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
Maria Reply
please solve
Sharon
8m/s²
Aishat
What is Thermodynamics
Muordit
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
Saheed Reply
50 m/s due south east
Someone
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
Ramon Reply
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
Scratch that
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
Someone
definitely of physics
Haryormhidey Reply
how many start and codon
Esrael Reply
what is field
Felix Reply
physics, biology and chemistry this is my Field
ALIYU
field is a region of space under the influence of some physical properties
Collete
what is ogarnic chemistry
WISDOM Reply
determine the slope giving that 3y+ 2x-14=0
WISDOM
Another formula for Acceleration
Belty Reply
a=v/t. a=f/m a
IHUMA
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Adah
pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?
Nassze Reply
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Esrael
Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.
JALLAH Reply
No. According to Isac Newtons law. this two bodies maybe you and the wall beside you. Attracting depends on the mass och each body and distance between them.
Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Robert
like charges repel while unlike charges atttact
Raymond
What is specific heat capacity
Destiny Reply
Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).
AI-Robot
specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin
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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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