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Mathematics LibreTexts

8.6: Adding and Subtracting Rational Expressions

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Basic Rule

We are now in a position to study the process of adding and subtracting rational expressions. There is a most basic rule to which we must strictly adhere if we wish to conveniently add or subtract rational expressions.

To add or subtract rational expressions conveniently, they should have the same denominators.

Thus, to add or subtract two or more rational expressions conveniently, we must ensure that they all have the same denominator. The denominator that is most convenient is the LCD.

Fractions With The Same Denominator

The Rule for Adding and Subtracting Rational Expressions

To add (or subtract) two or more rational expressions with the same denominators, add (or subtract) the numerators and place the result over the LCD. Reduce if necessary. Symbolically,

ac+bc=a+bc

acbc=abc

Note that we combine only the numerators.

Sample Set A

Add or subtract the following rational expressions.

Example 8.6.1

16+36 The denominators are the same. Add the numerators.16+36=1+36=46 Reduce 16+36=23

Example 8.6.2

5x+8x The denominators are the same. Add the numerators 5x+5+8x=13x

Example 8.6.3

2aby2w5by2w The denominators are the same. Subtract the numerators 2aby2w5by2w=2ab5by2w

Example 8.6.4

3x2+x+2x7+x24x+1x7 The denominators are the same. Add the numerators3x2+x+2x7+x24x+1x7=3x2+x+2+x24x+1x7=4x23x+3x7

Example 8.6.5

5y+32y52y+42y5 The denominators are the same. Subtract the numerators.  But be careful to subtract the entire numerator. Use parentheses!5y+32y52y+42y5=5y+3(2y+4)2y5=5y+32y42y5=3y12y5

 Note: 5y+32y52y+42y5
 The term 2y+42y5 could be written as
+(2y+4)2y5=2y42y5

Note

A common mistake is to write:

2y+42y5 as 2y+42y5

This is not correct, as the negative sign is not being applied to the entire numerator

Example 8.6.6

3x2+4x+5(x+6)(x2)+2x2+x+6x2+4x12x24x6x2+4x12

Factor the denominators to determine if they're the same:

3x2+4x+5(x+6)(x2)+2x2+x+6(x+6)(x2)x24x6(x+6)(x2)

The denominators are the same. Combine the numerators being careful to note the negative sign.

3x2+4x+5+2x2+x+6(x24x+6)(x+6)(x2)
3x2+4x+5+2x2+x+6x2+4x+6(x+6)(x2)
4x2+9x+17(x+6)(x2)

Practice Set A

Add or Subtract the following rational expressions.

Practice Problem 8.6.1

49+29

Answer

23

Practice Problem 8.6.2

3b+2b

Answer

5b

Practice Problem 8.6.3

5x2y23x2y2

Answer

xy2

Practice Problem 8.6.4

x+yxy+2x+3yxy

Answer

3x+4yxy

Practice Problem 8.6.5

4x2x+43x+10x2+2x+53x+10

Answer

3x23x13x+10

Practice Problem 8.6.6

x(x+1)x(2x+3)+3x2x+72x2+3x

Answer

4x2+7x(2x+3)

Practice Problem 8.6.7

4x+3x2x68x4(x+2)(x3)

Answer

4x+7(x+2)(x3)

Practice Problem 8.6.8

5a2+a42a(a6)+2a2+3a+42a212a+a2+22a212a

Answer

4a2+2a+1a(a6)

Practice Problem 8.6.9

8x2+x1x26x+8+2x2+3xx26x+85x2+3x4(x4)(x2)

Answer

5x2+x+3(x4)(x2)

Fractions with Different Denominators

Sample Set B

Add or Subtract the following rational expressions.

Example 8.6.7

4a3y+2a8y2.The denominators are not the same. Find the LCD. By inspection, the LCD is9y2The denominator of the first rational expression has been multiplied by 3y?9y2+2a9y2so the numerator must be multiplied by 3y4a3y=12ay12ay9y2+2a9y2The denominators are now the same. Add the numerators.12ay+2a9y2

Example 8.6.8

3bb+2+5bb3 The denominators are not the same. The LCD is (b+2)(b3)?(b+2)(b3)+?(b+2)(b3) The denominator of the first rational expression has been multiplied by b3, so the numerator must be multiplied by b3.3b(b3)3b(b3)(b+2)(b3)+?(b+2)(b3) The denominator of the second rational expression has been multiplied by b+2, so the numerator must be multiplied by b+2.5b(b+2).3b(b3)(b+2)(b3)+5b(b+2)(b+2)(b3) The denominators are now the same. Add the numerators.

3b(b3)+5b(b+2)(b3)(b+2)=3b29b+5b2+10b(b3)(b+2)=8b2+b(b3)(b+2)

Example 8.6.9

x+3x1+x24x+4 The denominators are not the same Find the LCDx+3x1+x24(x+1) The LCD is (x+1)(x1)?4(x+1)(x1)+?4(x+1)(x1) The denominator of the first rational expression has been multiplied by 4(x+1) the numerator must be multiplied by 4(x+1)4(x+3)(x+1)4(x+3)(x+1)4(x+1)(x1)+?4(x+1)(x1) The denominator of the second rational expression has been multiplied by (x1) so the numerator must be multiplied by x1(x1)(x2)4(x+3)(x+1)4(x+1)(x1)+(x1)(x2)4(x+1)(x2) The denominator are now the same. Add the numerators4(x+3)(x+1)+(x1)(x2)4(x+1)(x1)4(x2+4x+3)+x23x+24(x+1)(x1)
4x2+16x+12+x23x+24(x+1)(x1)=5x2+13x+144(x+1)(x1)

Example 8.6.10

x+5x27x+12+3x1x22x3Determine the LCDx+5(x4)(x3)+3x1(x3)(x+1)The LCD is (x4)(x3)(x+1)?(x4)(x3)(x+1)+?(x4)(x3)(x+1) The first numerator must be multipled by x+1 and the second by x4(x+5)(x+1)(x4)(x3)(x+1)+(3x1)(x4)(x4)(x3)(x+1) The denominators are now the same. Add the numerators (x+5)(x+1)+(3x1)(x4)(x4)(x3)(x+1)x2+6x+5=3x2+13x+4(x4)(x3)(x+1)4x27x+9(x4)(x3)(x+1)

Example 8.6.11

a+4a2+5a+6a4a25a24Determine the LCDa+4(a+3)(a+2)a4(a+3)(a8) The LCD is (a+3)(a+2)(a8)?(a+3)(a+2)(a8)?(a+3)(a+2)(a8) The first numerator must be multipled by a8 and the second by a+2.(a+4)(a8)(a+3)(a+2)(a8)(a4)(a+2)(a+3)(a+2)(a8) The denominators are now the same. Subtract the numerators.(a+4)(a8)(a4)(a+2)(a+3)(a+2)(a8)a24a32(a22a8)(a+3)(a+2)(a8)a24a32a2+2a+8(a+3)(a+2)(a8)2a24(a+3)(a+2)(a8) Factor 2 from the numerator.2(a+12)(a+3)(a+2)(a8)

Example 8.6.12

3x7x+5xx7 The denominators are nearly the same. They differ only in sign  Our technique is to factor 1 from one of them3x7x=3x(x7)=3xx73x7x+5xx7=3xx7+5xx7=3x+5xx7=2xx7

Practice Set B

Add or Subtract the following rational expressions.

Practice Problem 8.6.10

3x4a2+5x12a3

Answer

9ax+5x12a3

Practice Problem 8.6.11

5bb+1+3bb2

Answer

8b27b(b+1)(b2)

Practice Problem 8.6.12

a7a+2+a2a+3

Answer

2a24a25(a+2)(a+3)

Practice Problem 8.6.13

4x+1x+3x+5x3

Answer

3x219x18(x+3)(x3)

Practice Problem 8.6.14

2y3y+3y+1y+4

Answer

5y2+6y12y(y+4)

Practice Problem 8.6.15

a7a23a+2+a+2a26a+8

Answer

2a210a+26(a2)(a1)(a4)

Practice Problem 8.6.16

6b2+6b+92b2+4b+4

Answer

4b2+12b+6(b+3)2(b+2)2

Practice Problem 8.6.17

xx+4x23x3

Answer

2x25x+83(x+4)(x2)

Practice Problem 8.6.18

5x4x+7xx4

Answer

2xx4

Sample Set C

Combine the following rational expressions.

Example 8.6.13

3+7x1Rewrite the expression31+7x1The LCD is x13(x1)x1+7x1=3x3x1+7x1=3x3+7x1=3x+4x1

Example 8.6.14

3y+4y2y+3y6Rewrite the expression.3y+41y2y+3y6The LCD is y6(3y+4)(y6)y6yy+3y6=(3y+4)(y6)(y2y+3)y6=3y214y24y2+y3y6=2y213y27y6

Practice Set C

Practice Problem 8.6.19

Simplify 8+3x6

Answer

8x45x6

Practice Problem 8.6.20

Simplify 2a5a2+2a1a+3

Answer

a2a14a+3

Exercises

For the following problems, add or subtract the rational expressions.

Exercise 8.6.1

38+18

Answer

12

Exercise 8.6.2

19+49

Exercise 8.6.3

71025

Answer

310

Exercise 8.6.4

34512

Exercise 8.6.5

34x+54x

Answer

2x

Exercise 8.6.6

27y+37y

Exercise 8.6.7

6y5x+8y5x

Answer

14y5x

Exercise 8.6.8

9a7b+3a7b

Exercise 8.6.9

15n2m6n2m

Answer

9n2m

Exercise 8.6.10

8p11q3p11q

Exercise 8.6.11

y+4y6+y+8y6

Answer

2y+12y6

Exercise 8.6.12

y1y+4+y+7y+4

Exercise 8.6.13

a+6a1+3a+5a1

Answer

4a+11a1

Exercise 8.6.14

5a+1a+7+2a6a+7

Exercise 8.6.15

x+15x+x+35x

Answer

2x+45x

Exercise 8.6.16

a6a+2+a2a+2

Exercise 8.6.17

b+1b3+b+2b3

Answer

2b+3b3

Exercise 8.6.18

a+2a5a+3a5

Exercise 8.6.19

b+7b6b1b6

Answer

8b6

Exercise 8.6.20

2b+3b+1b4b+1

Exercise 8.6.21

3y+4y+82y5y+8

Answer

y+9y+8

Exercise 8.6.22

2a7a9+3a+5a9

Exercise 8.6.23

8x1x+215x+7x+2

Answer

7x8x+2

Exercise 8.6.24

72x2+16x3

Exercise 8.6.25

23x+46x2

Answer

2(x+1)3x2

Exercise 8.6.26

56y3218y5

Exercise 8.6.27

25a2110a3

Answer

4a110a3

Exercise 8.6.28

3x+1+5x2

Exercise 8.6.29

4x6+1x1

Answer

5(x2)(x6)(x1)

Exercise 8.6.30

2aa+13aa+4

Exercise 8.6.31

6yy+4+2yy+3

Answer

2y(4y+13)(y+4)(y+3)

Exercise 8.6.32

x1x3+x+4x4

Exercise 8.6.33

x+2x5+x1x+2

Answer

2x22x+9(x5)(x+2)

Exercise 8.6.34

a+3a3a+2a2

Exercise 8.6.35

y+1y1y+4y4

Answer

6y(y1)(y4)

Exercise 8.6.36

x1(x+2)(x3)+x+4x3

Exercise 8.6.37

y+2(y+1)(y+6)+y2y+6

Answer

y2(y+1)(y+6)

Exercise 8.6.38

2a+1(a+3)(a3)a+2a+3

Exercise 8.6.39

3a+5(a+4)(a1)2a1a1

Answer

2a24a+9(a+4)(a1)

Exercise 8.6.40

2xx23x+2+3x2

Exercise 8.6.41

4aa22a3+3a+1

Answer

7a9(a+1)(a3)

Exercise 8.6.42

3yy27y+12y2y3

Exercise 8.6.43

x1x2+6x+8+x+3x2+2x8

Answer

2(x2+x+4)(x+2)(x2)(x+4)

Exercise 8.6.44

a4a2+2a3+a+2a2+3a4

Exercise 8.6.45

x1x2+6x+8+x+3x2+2x8

Answer

2(x2+x+4)(x+2)(x2)(x+4)

Exercise 8.6.46

a4a2+2a3+a+2a2+3a4

Exercise 8.6.47

b3b2+9b+20+b+4b2+b12

Answer

2b2+3b+29(b3)(b+4)(b+5)

Exercise 8.6.48

y1y2+4y12y+3y2+6y16

Exercise 8.6.49

x+3x2+9x+13x5x24

Answer

x+29(x2)(x+2)(x+7)

Exercise 8.6.50

x1x24x+3+x+3x25x+6+2xx23x+2

Exercise 8.6.51

4xx2+6x+8+3x2+x6+x1x2+x12

Answer

5x43x334x2+34x60(x2)(x+2)(x3)(x+3)(x+4)

Exercise 8.6.52

y+2y21+y3y23y4y+3y25y+4

Exercise 8.6.53

a2a29a+18+a2a24a12a2a2a6

Answer

(a+5)(a2)(a+2)(a3)(a6)

Exercise 8.6.54

y2y2+6y+y+4y2+5y6

Exercise 8.6.55

a+1a3+3a2a+6a2a

Answer

a38a218a1a2(a+3)(a1)

Exercise 8.6.56

43b212b26b26b

Exercise 8.6.57

32x54x4+28x3+24x2

Answer

x3+2x2+6x+184x4(x2)(x+3)

Exercise 8.6.58

x+212x3+x+14x2+8x12x+316x232x+16

Exercise 8.6.59

2xx29x+14x212xx48x3

Answer

14x49x32x2+9x368x3(x+3)(x3)

Exercise 8.6.60

4+3x+2

Exercise 8.6.61

8+2x+6

Answer

8x+50x+6

Exercise 8.6.62

1+4x7

Exercise 8.6.63

3+5x6

Answer

3x13x6

Exercise 8.6.64

2+4xx+5

Exercise 8.6.65

1+3aa1

Answer

2a+1a1

Exercise 8.6.66

64yy+2

Exercise 8.6.67

2x+x24x+1

Answer

3x2+2x4x+1

Exercise 8.6.68

3y+4y2+2y5y+3

Exercise 8.6.69

x+2+x2+4x1

Answer

2x2+x+2x1

Exercise 8.6.70

b+6+2b+5b2

Exercise 8.6.71

3x1x48

Answer

5x+31x4

Exercise 8.6.72

4y+5y+19

Exercise 8.6.73

2y2+11y1y+43y

Answer

(y2+y+1)y+4

Exercise 8.6.74

5y22y+1y2+y62

Exercise 8.6.75

4a3+2a2+a1a2+11a+28+3a

Answer

7a3+35a2+85a1(a+7)(a+4)

Exercise 8.6.76

2x1x+6xx1

Exercise 8.6.77

5m6m+3mm6

Answer

2mm6

Exercise 8.6.78

a+783a+2a+13a8

Exercises For Review

Exercise 8.6.79

Simplify (x3y2z5)6(x2yz)2

Answer

x22y14z32

Exercise 8.6.80

Write 6a3b4c2a1b5c3 so that only positive exponents appear.

Exercise 8.6.81

Construct the graph of y=2x+4

An xy coordinate plane with gridlines, labeled negative five to five on both axes.

Answer

A graph of a line passing through three points with coordinates zero, four; one, two; and two, zero.

Exercise 8.6.82

Find the product x23x4x2+6x+5x2+5x+6x22x8

Exercise 8.6.83

Replace N with the proper quantity: x+3x5=Nx27x+10

Answer

(x+3)(x2)


This page titled 8.6: Adding and Subtracting Rational Expressions is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Denny Burzynski & Wade Ellis, Jr. (OpenStax CNX) .

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