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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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Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
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John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
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Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
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what are the types of wave
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answer
Magreth
progressive wave
Magreth
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hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
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Who can show me the full solution in this problem?
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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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