Lecture : Subtraction of Rational Expressions


 

In order to subtract fractions we must have a common denominator. Since rational expressions are in essence fractions, we must have a common denominator to subtract them. So before you go on be sure that you know how to find the Least Common Denominator (LCD) for two rational expressions.
 


    Click here to learn how to find the Least Common Denominator 

Once our rational expressions have the same denominator, we multiply and combine like terms in each numerators. Finally, we subtract the numerators, and put the answer over the LCD.

Here is an example:

 

  2x + 3 
x2 - 9

- 

    x + 4 
x2 + 5x + 6

First we must find a common denominator, so we factor both denominators

 

     2x + 3 
(x - 3)(x + 3)

- 

     x + 4 
(x + 2)(x + 3)

So the LCD = (x -3)(x + 3)(x + 2).

In our first rational expression we need to multiply the numerator and denominator by (x + 2) to get the LCD, and in the second fraction we must multiply the numerator and the denominator by (x - 3) to get the LCD:

 

     (2x + 3)(x+ 2) 
(x - 3)(x + 3)(x + 2)

- 

     (x + 4)(x - 3) 
(x + 2)(x + 3)(x + 2)

Now that our rational expressions have the same denominator we will multiply out the numerators and subtract them

 

2x2 + 4x + 3x + 6
(x - 3)(x + 3)(x + 2)

- 

x2 + 4x - 3x - 12
(x + 2)(x + 3)(x + 2)

= 

    2x2+ 7x + 6 
(x - 3)(x + 3)(x + 2)

- 

    x2 + x - 12 
(x + 2)(x + 3)(x + 2)

Note: When subtracting be sure to distribute the negative sign to each term in the second numerator

2x2 + 7x + 6 - x2 - x + 12  =  x2 + 6x + 18

our answer is:

     x2 + 6x + 18 
(x - 3)(x + 3)(x + 2)

We now need to check that this rational expression can not be simplified. Using the ac method of factoring, we will see that the numerator does not factor so the expression can not be further simplified.

Subtract each of the following.

1. 

       2 
x2 + 6x + 8

- 

       1 
x2 + 5x + 6

     :solution

2. 

       3x 
x2 - 2x - 15

- 

       2x 
2x2 + 5x - 3

     :solution

3. 

   x - 2 
x2 - 16

- 

       x - 2 
x2 - 8x + 16

     :solution

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#1 solution
 

       2 
x2 + 6x + 8

- 

       1 
x2 + 5x + 6

First we must find a common denominator, so we factor both denominators

 

       2 
(x + 4)(x + 2)

- 

       1 
(x + 3)(x + 2)

So the LCD = (x + 2)(x + 3)(x + 4).

In our first rational expression we need to multiply the numerator and denominator by (x + 3) to get the LCD, and in the second fraction we must multiply the numerator and the denominator by (x +4) to get the LCD:

 

       2(x + 3) 
(x + 4)(x + 3)(x + 2)

- 

       1(x + 4) 
(x + 4)(x + 3)(x + 2)

Now that our rational expressions have the same denominator we will multiply out the numerators and subtract them

 

          2x + 6 
(x + 4)(x + 3)(x + 2)

- 

          x + 4 
(x + 4)(x + 3)(x + 2)

Note: When subtracting be sure to distribute the negative sign to each term in the second numerator

2x + 6 - x - 4   =  x + 2

So we get:

          x + 2 
(x + 4)(x + 3)(x + 2)

This can be reduced since x + 2 is common to both the numerator and denominator, after cancelling the common factor of (x + 2) we get the following answer.
 

          1 
(x + 4)(x + 3)

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#2 solution
 

       3x 
x2 -2x - 15

- 

       2x 
2x2 + 5x - 3

First we must find a common denominator, so we factor both denominators

 

       3x 
(x - 5)(x + 3)

- 

       2x 
(2x - 1)(x + 3)

 

So the LCD = (x - 5)(x + 3)(2x - 1)

In our first rational expression we need to multiply the numerator and denominator by (2x - 1) to get the LCD, and in the second fraction we must multiply the numerator and the denominator by (x - 5) to get the LCD:

 

       3x(2x - 1) 
(x - 5)(x + 3)(2x - 1)

- 

       2x(x - 5) 
(x - 5)(x + 3)(2x - 1)

Now that our rational expressions have the same denominator we will multiply out the numerators and subtract them

 

       6x2 - 3x 
(x - 5)(x + 3)(2x - 1)

- 

       2x2 - 10x 
(x - 5)(x + 3)(2x - 1)

When subtracting be sure to distribute the negative sign to each term in the second numerator

6x2 + 3x - 2x2 + 10x   =  4x2 + 13x

and we get:

       4x2 + 13x 
(x - 5)(x + 3)(2x - 1)

We now need to check that this rational expression can not be simplified. We factor the numerator, to see if we have any common factors.

 

       x(4x + 13) 
(x - 5)(x + 3)(2x - 1)

Since we have no common factors in the numerator and the denominator, this is our answer.

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#3 solution

   x - 2 
x2 - 16

- 

       x - 2 
x2 - 8x + 16

First we must find a common denominator, so we factor both denominators

 

      x - 2 
(x - 4)(x + 4)

- 

       x - 2 
(x - 4)(x - 4)

So the LCD = (x + 4)( x - 4)(x - 4) or (x - 4)2(x + 4)

In our first rational expression we need to multiply the numerator and denominator by (x - 4) to get the LCD, and in the second fraction we must multiply the numerator and the denominator by (x + 4) to get the LCD:

 

  (x - 2)(x - 4) 
(x - 4)2(x + 4)

- 

  (x - 2)(x + 4) 
(x - 4)2(x + 4)

Now that our rational expressions have the same denominator we will multiply out the numerators and subtract them

 

x2 – 4x - 2x + 8
(x - 4)2(x + 4)

- 

x2 + 4x - 2x - 8
(x - 4)2(x + 4)

= 

x2 - 6x + 8
(x - 4)2(x + 4)

- 

x2+ 2x - 8
(x - 4)2(x + 4)

When subtracting be sure to distribute the negative sign to each term in the second numerator

x2 - 6x + 8 - x2 - 2x + 8    =   - 8x + 16

So we get:

    -8x + 16 
(x - 4)2(x + 4)

We now need to check that this rational expression can not be simplified. We factor the numerator, to see if we have any common factors.

 

   -8(x - 2) 
(x - 4)2(x + 4)

Since we have no common factors in the numerator and denominator, this is our answer.

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