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3 x 2 + 4 x + 5 ( x + 6 ) ( x 2 ) + 2 x 2 + x + 6 x 2 + 4 x 12 x 2 4 x 6 x 2 + 4 x 12 Factor the denominators to determine if they're the same . 3 x 2 + 4 x + 5 ( x + 6 ) ( x 2 ) + 2 x 2 + x + 6 ( x + 6 ) ( x 2 ) x 2 4 x 6 ( x + 6 ) ( x 2 ) The denominators are the same . Combine the numerators being careful to note the negative sign . 3 x 2 + 4 x + 5 + 2 x 2 + x + 6 ( x 2 4 x + 6 ) ( x + 6 ) ( x 2 ) 3 x 2 + 4 x + 5 + 2 x 2 + x + 6 x 2 + 4 x + 6 ( x + 6 ) ( x 2 ) 4 x 2 + 9 x + 17 ( x + 6 ) ( x 2 )

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Practice set a

Add or Subtract the following rational expressions.

5 x 2 y 2 3 x 2 y 2

x y 2

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x + y x y + 2 x + 3 y x y

3 x + 4 y x y

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4 x 2 x + 4 3 x + 10 x 2 + 2 x + 5 3 x + 10

3 x 2 3 x 1 3 x + 10

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x ( x + 1 ) x ( 2 x + 3 ) + 3 x 2 x + 7 2 x 2 + 3 x

4 x 2 + 7 x ( 2 x + 3 )

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4 x + 3 x 2 x 6 8 x 4 ( x + 2 ) ( x 3 )

4 x + 7 ( x + 2 ) ( x 3 )

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5 a 2 + a 4 2 a ( a 6 ) + 2 a 2 + 3 a + 4 2 a 2 12 a + a 2 + 2 2 a 2 12 a

4 a 2 + 2 a + 1 a ( a 6 )

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8 x 2 + x 1 x 2 6 x + 8 + 2 x 2 + 3 x x 2 6 x + 8 5 x 2 + 3 x 4 ( x 4 ) ( x 2 )

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

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Fractions with different denominators

Sample set b

Add or Subtract the following rational expressions.

4 a 3 y + 2 a 9 y 2 . The denominators are  n o t  the same . Find the LCD . By inspection, the LCD is 9 y 2 . 9 y 2 + 2 a 9 y 2 The denominator of the first rational expression has been multiplied by 3 y , so the numerator must be multiplied by 3 y . 4 a · 3 y = 12 a y 12 a y 9 y 2 + 2 a 9 y 2 The denominators are now the same . Add the numerators . 12 a y + 2 a 9 y 2

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3 b b + 2 + 5 b b 3 . The denominators are n o t the same . The LCD is ( b + 2 ) ( b 3 ) . ( b + 2 ) ( b 3 ) + ( b + 2 ) ( b 3 ) The denominator of the first rational expression has been multiplied by b 3 , so the numerator must be multiplied by b 3. 3 b ( b 3 ) 3 b ( b 3 ) ( b + 2 ) ( b 3 ) + ( b + 2 ) ( b 3 ) The denominator of the second rational expression has been multiplied by b + 2 , so the numerator must be multiplied by b + 2. 5 b ( b + 2 ) 3 b ( b 3 ) ( b + 2 ) ( b 3 ) + 5 b ( b + 2 ) ( b + 2 ) ( b 3 ) The denominators are now the same . Add the numerators . 3 b ( b 3 ) + 5 b ( b + 2 ) ( b 3 ) ( b + 2 ) = 3 b 2 9 b + 5 b 2 + 10 b ( b 3 ) ( b + 2 ) = 8 b 2 + b ( b 3 ) ( b 2 )

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x + 3 x 1 + x 2 4 x + 4 . The denominators are  n o t  the same . Find the LCD . x + 3 x 1 + x 2 4 ( x + 1 ) The LCD is  ( x + 1 ) ( x 1 ) 4 ( x + 1 ) ( x 1 ) + 4 ( x + 1 ) ( x 1 ) The denominator of the first rational expression has been multiplied by  4 ( x + 1 )  so  the numerator must be multiplied by  4 ( x + 1 ) . 4 ( x + 3 ) ( x + 1 ) 4 ( x + 3 ) ( x + 1 ) 4 ( x + 1 ) ( x 1 ) + 4 ( x + 1 ) ( x 1 ) The denominator of the second rational expression has been multiplied by  x 1 so the numerator must be multiplied by  x 1. ( x 1 ) ( x 2 ) 4 ( x + 3 ) ( x + 1 ) 4 ( x + 1 ) ( x 1 ) + ( x 1 ) ( x 2 ) 4 ( x + 1 ) ( x 1 ) The denominators are now the same. Add the numerators. 4 ( x + 3 ) ( x + 1 ) + ( x 1 ) ( x 2 ) 4 ( x + 1 ) ( x 1 ) 4 ( x 2 + 4 x + 3 ) + x 2 3 x + 2 4 ( x + 1 ) ( x 1 ) 4 x 2 + 16 x + 12 + x 2 3 x + 2 4 ( x + 1 ) ( x 1 ) = 5 x 2 + 13 x + 14 4 ( x + 1 ) ( x 1 )

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x + 5 x 2 7 x + 12 + 3 x 1 x 2 2 x 3 Determine the LCD . x + 5 ( x 4 ) ( x 3 ) + 3 x 1 ( x 3 ) ( x + 1 ) The LCD is ( x 4 ) ( x 3 ) ( x + 1 ) . ( x 4 ) ( x 3 ) ( x + 1 ) + ( x 4 ) ( x 3 ) ( x + 1 ) The first numerator must be multiplied by x + 1 and the second by x 4. ( x + 5 ) ( x + 1 ) ( x 4 ) ( x 3 ) ( x + 1 ) + ( 3 x 1 ) ( x 4 ) ( x 4 ) ( x 3 ) ( x + 1 ) The denominators are now the same . Add the numerators . ( x + 5 ) ( x + 1 ) + ( 3 x 1 ) ( x 4 ) ( x 4 ) ( x 3 ) ( x + 1 ) x 2 + 6 x + 5 + 3 x 2 13 x + 4 ( x 4 ) ( x 3 ) ( x + 1 ) 4 x 2 7 x + 9 ( x 4 ) ( x 3 ) ( x + 1 )

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a + 4 a 2 + 5 a + 6 a 4 a 2 5 a 24 Determine the LCD. a + 4 ( a + 3 ) ( a + 2 ) a 4 ( a + 3 ) ( a 8 ) The LCD is  ( a + 3 ) ( a + 2 ) ( a 8 ) . ( a + 3 ) ( a + 2 ) ( a 8 ) ( a + 3 ) ( a + 2 ) ( a 8 ) The first numerator must be multiplied by  a 8  and the second by  a + 2. ( a + 4 ) ( a 8 ) ( a + 3 ) ( a + 2 ) ( a 8 ) ( a 4 ) ( a + 2 ) ( a + 3 ) ( a + 2 ) ( a 8 ) The denominators are now the same . Subtract the numerators .  ( a + 4 ) ( a 8 ) ( a 4 ) ( a + 2 ) ( a + 3 ) ( a + 2 ) ( a 8 ) a 2 4 a 32 ( a 2 2 a 8 ) ( a + 3 ) ( a + 2 ) ( a 8 ) a 2 4 a 32 a 2 + 2 a + 8 ( a + 3 ) ( a + 2 ) ( a 8 ) 2 a 24 ( a + 3 ) ( a + 2 ) ( a 8 ) Factor  2  from the numerator .  2 ( a + 12 ) ( a + 3 ) ( a + 2 ) ( a 8 )

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3 x 7 x + 5 x x 7 . The denominators are n e a r l y the same.  They differ only in sign. Our technique is to factor  1  from one of them. 3 x 7 x = 3 x ( x 7 ) = 3 x x 7 Factor  1  from the first term .  3 x 7 x + 5 x x 7 = 3 x x 7 + 5 x x 7 = 3 x + 5 x x 7 = 2 x x 7

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Practice set b

Add or Subtract the following rational expressions.

3 x 4 a 2 + 5 x 12 a 3

9 a x + 5 x 12 a 3

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5 b b + 1 + 3 b b 2

8 b 2 7 b ( b + 1 ) ( b 2 )

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a 7 a + 2 + a 2 a + 3

2 a 2 4 a 25 ( a + 2 ) ( a + 3 )

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4 x + 1 x + 3 x + 5 x 3

3 x 2 19 x 18 ( x + 3 ) ( x 3 )

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2 y 3 y + 3 y + 1 y + 4

5 y 2 + 6 y 12 y ( y + 4 )

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a 7 a 2 3 a + 2 + a + 2 a 2 6 a + 8

2 a 2 10 a + 26 ( a 2 ) ( a 1 ) ( a 4 )

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6 b 2 + 6 b + 9 2 b 2 + 4 b + 4

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

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x x + 4 x 2 3 x 3

2 x 2 5 x + 8 3 ( x + 4 ) ( x 1 )

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5 x 4 x + 7 x x 4

2 x x 4

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Sample set c

Combine the following rational expressions.

3 + 7 x 1 . Rewrite the expression. 3 1 + 7 x 1 The LCD is  x 1. 3 ( x 1 ) x 1 + 7 x 1 = 3 x 3 x 1 + 7 x 1 = 3 x 3 + 7 x 1 = 3 x + 4 x 1

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3 y + 4 y 2 y + 3 y 6 . Rewrite the expression. 3 y + 4 1 y 2 y + 3 y 6 The LCD is  y 6. ( 3 y + 4 ) ( y 6 ) y 6 y 2 y + 3 y 6 = ( 3 y + 4 ) ( y 6 ) ( y 2 y + 3 ) y 6 = 3 y 2 14 y 24 y 2 + y 3 y 6 = 2 y 2 13 y 27 y 6

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Practice set c

Simplify 8 + 3 x 6 .

8 x 45 x 6

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Simplify 2 a 5 a 2 + 2 a 1 a + 3 .

a 2 a 14 a + 3

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Exercises

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

6 y 5 x + 8 y 5 x

14 y 5 x

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15 n 2 m 6 n 2 m

9 n 2 m

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y + 4 y 6 + y + 8 y 6

2 y + 12 y 6

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y 1 y + 4 + y + 7 y + 4

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a + 6 a 1 + 3 a + 5 a 1

4 a + 11 a 1

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5 a + 1 a + 7 + 2 a 6 a + 7

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x + 1 5 x + x + 3 5 x

2 x + 4 5 x

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a 6 a + 2 + a 2 a + 2

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b + 1 b 3 + b + 2 b 3

2 b + 3 b 3

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a + 2 a 5 a + 3 a 5

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b + 7 b 6 b 1 b 6

8 b 6

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2 b + 3 b + 1 b 4 b + 1

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3 y + 4 y + 8 2 y 5 y + 8

y + 9 y + 8

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2 a 7 a 9 + 3 a + 5 a 9

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8 x 1 x + 2 15 x + 7 x + 2

7 x 8 x + 2

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2 3 x + 4 6 x 2

2 ( x + 1 ) 3 x 2

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2 5 a 2 1 10 a 3

4 a 1 10 a 3

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4 x 6 + 1 x 1

5 ( x 2 ) ( x 6 ) ( x 1 )

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6 y y + 4 + 2 y y + 3

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

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x 1 x 3 + x + 4 x 4

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x + 2 x 5 + x 1 x + 2

2 x 2 2 x + 9 ( x 5 ) ( x + 2 )

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a + 3 a 3 a + 2 a 2

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y + 1 y 1 y + 4 y 4

6 y ( y 1 ) ( y 4 )

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

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

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

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

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3 a + 5 ( a + 4 ) ( a 1 ) 2 a 1 a 1

2 a 2 4 a + 9 ( a + 4 ) ( a 1 )

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2 x x 2 3 x + 2 + 3 x 2

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4 a a 2 2 a 3 + 3 a + 1

7 a 9 ( a + 1 ) ( a 3 )

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3 y y 2 7 y + 12 y 2 y 3

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x 1 x 2 + 6 x + 8 + x + 3 x 2 + 2 x 8

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

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a 4 a 2 + 2 a 3 + a + 2 a 2 + 3 a 4

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b 3 b 2 + 9 b + 20 + b + 4 b 2 + b 12

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

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y 1 y 2 + 4 y 12 y + 3 y 2 + 6 y 16

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x + 3 x 2 + 9 x + 14 x 5 x 2 4

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

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x 1 x 2 4 x + 3 + x + 3 x 2 5 x + 6 + 2 x x 2 3 x + 2

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4 x x 2 + 6 x + 8 + 3 x 2 + x 6 + x 1 x 2 + x 12

5 x 4 3 x 3 34 x 2 + 34 x 60 ( x 2 ) ( x + 2 ) ( x 3 ) ( x + 3 ) ( x + 4 )

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y + 2 y 2 1 + y 3 y 2 3 y 4 y + 3 y 2 5 y + 4

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a 2 a 2 9 a + 18 + a 2 a 2 4 a 12 a 2 a 2 a 6

( a + 5 ) ( a 2 ) ( a + 2 ) ( a 3 ) ( a 6 )

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y 2 y 2 + 6 y + y + 4 y 2 + 5 y 6

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a + 1 a 3 + 3 a 2 a + 6 a 2 a

a 3 8 a 2 18 a 1 a 2 ( a + 3 ) ( a 1 )

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4 3 b 2 12 b 2 6 b 2 6 b

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3 2 x 5 4 x 4 + 2 8 x 3 + 24 x 2

x 3 + 2 x 2 + 6 x + 18 4 x 4 ( x 2 ) ( x + 3 )

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x + 2 12 x 3 + x + 1 4 x 2 + 8 x 12 x + 3 16 x 2 32 x + 16

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2 x x 2 9 x + 1 4 x 2 12 x x 4 8 x 3

14 x 4 9 x 3 2 x 2 + 9 x 36 8 x 3 ( x + 3 ) ( x 3 )

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8 + 2 x + 6

8 x + 50 x + 6

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3 + 5 x 6

3 x 13 x 6

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1 + 3 a a 1

2 a + 1 a 1

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2 x + x 2 4 x + 1

3 x 2 + 2 x 4 x + 1

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3 y + 4 y 2 + 2 y 5 y + 3

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x + 2 + x 2 + 4 x 1

2 x 2 + x + 2 x 1

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3 x 1 x 4 8

5 x + 31 x 4

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2 y 2 + 11 y 1 y + 4 3 y

( y 2 + y + 1 ) y + 4

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5 y 2 2 y + 1 y 2 + y 6 2

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4 a 3 + 2 a 2 + a 1 a 2 + 11 a + 28 + 3 a

7 a 3 + 35 a 2 + 85 a 1 ( a + 7 ) ( a + 4 )

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2 x 1 x + 6 x x 1

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5 m 6 m + 3 m m 6

2 m m 6

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a + 7 8 3 a + 2 a + 1 3 a 8

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2 y + 4 4 5 y 9 5 y 4

2 y 13 5 y 4

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m 1 1 m 2 m 1

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Exercises for review

( [link] ) Simplify ( x 3 y 2 z 5 ) 6 ( x 2 y z ) 2 .

x 22 y 14 z 32

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( [link] ) Write 6 a 3 b 4 c 2 a 1 b 5 c 3 so that only positive exponents appear.

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( [link] ) Construct the graph of y = 2 x + 4.
An xy coordinate plane with gridlines, labeled negative five to five on both axes.

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

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( [link] ) Find the product: x 2 3 x 4 x 2 + 6 x + 5 · x 2 + 5 x + 6 x 2 2 x 8 .

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( [link] ) Replace N with the proper quantity: x + 3 x 5 = N x 2 7 x + 10 .

( x + 3 ) ( x 2 )

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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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