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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

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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