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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

what is the VA Ha D R X int Y int of f(x) =x²+4x+4/x+2 f(x) =x³-1/x-1
Shadow Reply
can I get help with this?
Wayne
Are they two separate problems or are the two functions a system?
Wilson
Also, is the first x squared in "x+4x+4"
Wilson
x^2+4x+4?
Wilson
thank you
Wilson
Please see ***imgur.com/a/lpTpDZk for solutions
Wilson
f(x)=x square-root 2 +2x+1 how to solve this value
Marjun Reply
factor or use quadratic formula
Wilson
what is algebra
Ige Reply
The product of two is 32. Find a function that represents the sum of their squares.
Paul
if theta =30degree so COS2 theta = 1- 10 square theta upon 1 + tan squared theta
Martin Reply
how to compute this 1. g(1-x) 2. f(x-2) 3. g (-x-/5) 4. f (x)- g (x)
Yanah Reply
hi
John
hi
Grace
what sup friend
John
not much For functions, there are two conditions for a function to be the inverse function:   1--- g(f(x)) = x for all x in the domain of f     2---f(g(x)) = x for all x in the domain of g Notice in both cases you will get back to the  element that you started with, namely, x.
Grace
sin theta=3/4.prove that sec square theta barabar 1 + tan square theta by cosec square theta minus cos square theta
Umesh Reply
acha se dhek ke bata sin theta ke value
Ajay
sin theta ke ja gha sin square theta hoga
Ajay
I want to know trigonometry but I can't understand it anyone who can help
Siyabonga Reply
Yh
Idowu
which part of trig?
Nyemba
functions
Siyabonga
trigonometry
Ganapathi
differentiation doubhts
Ganapathi
hi
Ganapathi
hello
Brittany
Prove that 4sin50-3tan 50=1
Sudip Reply
False statement so you cannot prove it
Wilson
f(x)= 1 x    f(x)=1x  is shifted down 4 units and to the right 3 units.
Sebit Reply
f (x) = −3x + 5 and g (x) = x − 5 /−3
Sebit
what are real numbers
Marty Reply
I want to know partial fraction Decomposition.
Adama Reply
classes of function in mathematics
Yazidu Reply
divide y2_8y2+5y2/y2
Sumanth Reply
wish i knew calculus to understand what's going on 🙂
Dashawn Reply
@dashawn ... in simple terms, a derivative is the tangent line of the function. which gives the rate of change at that instant. to calculate. given f(x)==ax^n. then f'(x)=n*ax^n-1 . hope that help.
Christopher
thanks bro
Dashawn
maybe when i start calculus in a few months i won't be that lost 😎
Dashawn
what's the derivative of 4x^6
Axmed Reply
24x^5
James
10x
Axmed
24X^5
Taieb
comment écrire les symboles de math par un clavier normal
SLIMANE
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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