When adding and subtracting fractions, it is necessary to find a common denominator – preferably the lowest common denominator (LCD). This is what makes addition and subtraction of fractions more tedious than multiplication and division. A common denominator is NOT required for multiplication or division. The process of determining the LCD is critical when adding or subtracting fractions but before dealing with examples where the denominators are different, it might be useful to work with some where the denominators are the same. In these cases, it will not be necessary to determine the LCD because the denominators will start out that way.

When the denominators of the fractions being added or subtracted start out the same, it is necessary to combine the numerators over that same denominator. It may be possible to factor the denominator and the newly combined numerator in a later step.

x x – 8 2x 2 – 1x + 16x – 8 x(2x – 1) + 8(2x – 1) (2x – 1)(x + 8) 2 × - 8 = x x – 8 3x 2 + 2x - 12x – 8 x(3x + 2) - 4(3x + 2) (3x + 2)(x - 4) 3 × - 8 =

When adding and subtracting fractions where the denominators are different, it is necessary to find the LCD. This is what makes addition and subtraction of fractions more tedious than multiplication and division. A common denominator is NOT required for multiplication or division. The LCD is basically the simplest expression that all of the starting denominators can divide into. If the denominators are just numerical, then the LCD is actually the lowest common multiple of each of the original denominators. Multiples of 5 = 5, 10, 15, 20, 25, 30, 35, 45, 50,55, … Multiples of 9 = 9, 18, 27, 36, 45, 54,63, …  LCM = 45 Multiples of 18 = 18, 36, 54, 72, 90, 108, … Multiples of 12 = 12, 24, 36, 48, 60, 72, 84, …  LCM = 36

To add these 2 fractions, we must find the lowest multiple of denominators 6 and 8 – in other words ‘the lowest common denominator’. LCD = 24 The steps involved in the addition of algebraic fractions can be appreciated by analyzing the steps of adding of a simpler pair of numeric fractions. The LCD for these 2 fractions is 24 because 24 is the lowest number that both 6 and 8 can divide into evenly. 24  8 = 3 and 24  6 = 4 After choosing the LCD, it is now necessary to rewrite each of the fractions with the LCD instead of the original denominators. NOTE: If the denominator is changed then the numerator must be changed by the same factor to preserve the value of the original fractions. 9 20

Once the fractions have been written with a common denominator, the operation can be completed in the numerator. This whole process basically has 3 critical steps: 1.Choosing the LCD. - Technically any common denominator will yield the correct answer (not just the lowest), however if a greater denominator is used it causes all the numbers thereafter to be bigger and therefore more difficult to work with. When working with algebraic fractions a common denominator that is NOT as small as possible will often be crippling and make the expressions too complicated to work with. 2.Adjusting the numerator of each fraction. - When the denominator of a fraction is changed to the LCD, the numerator must be changed by the same factor. 3.Performing operations in numerator. - Only the numerators are added/subtracted. The denominator remains whatever the LCD is.

When adding or subtracting fractions with different denominators, we will deal with them first by working with fractions that have monomial (one-term) denominators. LCD = 6 15x 14x 7x LCD = 12 8x

To determine the LCD for monomial expressions involving variables, it is helpful to visualize the basic or prime factors of each of the denominators of the fractions being added or subtracted. 6x 2 y 3 z = 2  3  x  x  y  y  y  z 9xy 3 z 2 = 3  3  x  y  y  y  z  z  8 basic factors  8 basic factors The LCD must contain all of those basic factors. If a factor exists in one of the original denominators more than once, then it must also be in the LCD more than once. However, if a factor is in both denominators only once, then it only needs to be in the LCD once.

6x 2 y 3 z = 2  3  x  x  y  y  y  z 9xy 3 z 2 = 3  3  x  y  y  y  z  z  8 basic factors  8 basic factors LCD = 2  3 Referring to the example, the LCD must contain a factor of 2 and 3 because these are factors of 6x 2 y 3 z.  3 It must contain a second factor of 3 because there are two factors of 3 in 9xy 3 z 2. It must contain two factors of x because there are two factors of x in 6x 2 y 3 z. 9xy 3 z 2 has only 1 factor of x and it is already there once.  x  x It must contain a three factors of y because there are 3 factors of y in 6x 2 y 3 z. 9xy 3 z 2 also has 3 factors of y but it is already part of the LCD 3 times.  y  y  y It must contain a factor of z because there is a factor of z in 6x 2 y 3 z. 9xy 3 z 2 has 2 factors of z. The LCD already has been given one but it must be given a second.  z  z= 18x 2 y 3 z 2 The LCD has 10 basic factors.

Multiples of 6 = 6, 12, 18, 24, 30, 36, 42, … Multiples of 9 = 9, 18, 27, 36, 45, 54, … Another way to determine the LCD is to determine the numeric part of the LCD by choosing the lowest common multiple and then determine the variable part by choosing the greatest exponent from each of the variables of the original denominators. 6x 2 y 3 z9xy 3 z 2 LCM = 18 The LCD for the x variable will be x 2 because 2 is the greatest exponent for the x variable. The LCD for the y variable will be y 3 because 3 is the greatest exponent for the y variable. The LCD for the z variable will be z 2 because 2 is the greatest exponent for the z variable. LCM = 18x 2 y 3 z 2

LCD = xy 5y 7x LCD = x 2 5x 3

LCD = 30x 3 y 3 25x 9y 2 LCD = ab 2 c 2 d 2 a 3 bd c3 c3

To determine the LCD for denominators having expressions involving 2 or more terms, it is again helpful to visualize the basic or prime factors of each of the denominators of the fractions being added or subtracted. For this reason, it is necessary to factor the denominators to their most basic factors at the very beginning. There are 2 factors in each denominator. The LCD must contain those factors. LCD = (x + 2)(x + 4)(x – 2) The first denominator has the factors (x + 2) and (x + 4) so they must be a part of the LCD. The second denominator has the factors (x + 2) and (x - 2) so they must be a part of the LCD. Since (x + 2) already is there, it doesn’t need to be placed there a second time.

There are 6 factors in the first denominator and 5 factors in the second. The LCD must contain those factors. LCD = 2  3  x  x(x + 2)(x + 4)3(x + 2) The first denominator has the factors 2, 3, x, x a second time, (x + 2) and (x + 4) so they must be a part of the LCD. The second denominator has the factors 3, 3 a second time, x, (x + 2) and (x + 2) a second time so they must be a part of the LCD. Since 3, x, and (x + 2) are already there one time, they don’t need to be placed there again. However because 3 and (x + 2) are in the second denominator twice, they must be a part of the LCD twice. = 2  3  3  x  x (x + 2)(x + 2)(x + 4) = 18x 2 (x + 2) 2 (x + 4)

LCD = 18x 2 (x + 2) 2 (x + 4) 21(x + 2) 28x(x + 4)

LCD = 18x 2 (x + 2) 2 (x + 4) 21(x + 2) 28x(x + 4) Be careful with subtraction of fractions.

LCD = (x + 3)(x + 4)(x – 2) 1(x – 2) 5(x + 4)