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Adding and Subtracting Rational
Expressions
Goal 1
Goal 2
Determine the LCM of
polynomials
Add and Subtract Rational
Expressions
What is the Least Common Multiple?
Fractions require you to find the Least Common Multiple (LCM)
in order to add and subtract them!
Least Common Multiple (LCM) - smallest number or
polynomial into which each of the numbers or
polynomials will divide evenly.
The Least Common Denominator is the LCM of
the denominators.
Find the LCM of each set of Polynomials
1) 12y2, 6x2
LCM = 12x2y2
2) 16ab3, 5a2b2, 20ac
3) x2 – 2x, x2 - 4
LCM = 80a2b3c
LCM = x(x + 2)(x – 2)
4) x2 – x – 20, x2 + 6x + 8
LCM = (x + 4) (x – 5) (x + 2)
Adding Fractions - A Review
9
8
3 2

 
4 3 12 12
9+8
=
12
17
=
12
LCD is 12.
Find equivalent
fractions using the
LCD.
Collect the numerators,
keeping the LCD.
Remember: When adding or subtracting
fractions, you need a common
denominator!
3 1 4
a.  
5 5 5
When Multiplying
or Dividing
Fractions, you
c.
don’t need a
common
Denominator
4 3
2 1
1
b.   

6 6
3 2
6
1
2
3
4
1 4
4
2
 


2 3
6
3
Steps for Adding and Subtracting Rational Expressions:
1. Factor
2. If possible, Cancel common factors directly over each
other.
3. Get a common denominator
• List out factors
• Multiply by 1’s
4. Add the top (combine like terms)
5. Re-factor and simplify if possible
Addition and Subtraction
Is the denominator the same??
Find the LCD: 6x
• Example: Simplify 2  5
3x
2x
Now, rewrite the expression using the LCD of 6x
2 2  5 3




3x 2
2x 3 Simplify...
4
15


6x 6x
4  15
Add the fractions...

6x
= 19
6x
Example 1
Simplify:
6
8
7


2
2
5m 3m n mn

6(3mn2) + 8(5n) - 7(15m)
2
15m n
2
Adding and Subtracting with polynomials as denominators
Simplify:
3
8

x2 x2
Find the LCD: (x + 2)(x – 2)
Rewrite the expression using the LCD of (x + 2)(x – 2)
3  x  2 
8 x  2 




(x  2) x  2
(x  2) x  2 
3x  6
8x  16


x  2x  2 x  2x  2
3x  6  (8x  16)

(x  2)(x  2)
Simplify...
3x  6  8x  16

(x  2)(x  2)
– 5x – 22
(x + 2)(x – 2)
Adding and Subtracting with Binomial Denominators
2
3

x 3 x1
2 (x + 1)

+ 3 (x + 3)
(x  3)(x  1)
Example 6
Simplify:
4
x
 3
3
3x 6 x  3x 2
4
x
 3 2
3x 3x (2 x  1)
** Needs a common denominator 1st!
Sometimes it helps to factor the
denominators to make it easier to find
your LCD.
LCD: 3x3(2x+1)
4(2 x  1)
x
 3
 3
3x (2 x  1) 3x (2 x  1)
2
4(2 x  1)  x

3x 3 (2 x  1)
2
x 2  8x  4
 3
3x (2 x  1)
Example 7
Simplify:
x 1
1
 2
2
x 6 x  9 x  9
x 1
1


( x  3)( x  3) ( x  3)( x  3)
LCD: (x+3)2(x-3)
( x  1)( x  3)
( x  3)
( x  1)( x  3)  ( x  3)



2
2
( x  3) ( x  3) ( x  3) ( x  3)
( x  3) 2 ( x  3)
x  3x  x  3  x  3

2
( x  3) ( x  3)
2
x 2  3x  6

( x  3) 2 ( x  3)
Example 9
Simplify:
3x
2x
3x (x - 1) - 2x (x + 2)
 2

2
x  5x  6 x  2x  3 (x  3)(x  2)(x  1)
(x + 3)(x + 2)
(x + 3)(x - 1)
LCD
(x + 3)(x + 2)(x - 1)
x ≠ -3, -2, 1
3x  3x  2x  4x

(x  3)(x  2)(x  1)
2
2
x  7x

(x  3)(x  2)(x  1)
2
Example 10
Simplify:
4x
5x
4x(x - 2) + 5x (x - 3)
 2

2
x  5x  6 x  4x  4
(x  3)(x  2)(x  2)
(x - 3)(x - 2) (x - 2)(x - 2)
LCD
(x - 3)(x - 2)(x - 2)
x ≠ 3, 2
4x 2  8x  5x 2 15x

(x  3)(x  2)(x  2)
9x  23x

(x  3)(x  2)(x  2)
2
Example 11
Simplify:
x3
x4
(x + 3) (x - 2) - (x - 4)(x + 1)

 2
2
(x  1)(x  1)(x  2)
x  1 x  3x  2
(x - 1)(x + 1)
(x - 2)(x - 1)
LCD
(x - 1)(x + 1)(x - 2)
x ≠ 1, -1, 2
(x 2  x  6)  (x2  3x  4)

(x 1)(x 1)(x  2)
4x  2

(x  1)(x  1)(x  2)