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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

write and explain four psychosexual stages of personality development
MPHID Reply
I'm asking on behalf of someone How can a guy get over his weakness for girls? Like the person have soft spot for every girl and he thinks he love them then suddenly stuff change.... Like a switch
Ajayi Reply
yes
Beverly
whats the age of the boy?
Beverly
what are cluster A disease?
Saee Reply
because psychology is a natural science as well as a science. it's a interdisciplinary subject
Kamakshi Reply
yes absolutely we cabt out Psychology completely into science dur to its various measuring aspects
utkarsh
cannot* put*
utkarsh
Assess the relationship between heredity and environment in human development
Newman Reply
जिवन कक्षा विकासाच्या एकूण किती अवस्था आहेत
Digambar Reply
what happened Digambar?
sakshi
?
Reina
Is here anybody exactly awared about the medicine used for the schizophrenia?
Fazil
plz can i know if io psychology in demand ou it's better to continu in human resourses management?
Reina
there are many drugs used to alleviate the symptoms, but none are used for curing the disorder itself.
Rupert
Researchers believe that one important function of sleep is to facilitate learning and memory. How does knowing this help you in your college studies? What changes could you make to your study and sleep habits to maximize your mastery of the material covered in class?
whitecat Reply
Well, for starters, both are entirely different aspect for consideration. As a college student, having an adequate amount of sleep is relatively rare especially on instances where we are totally forced to stay all night (whether for studying or doing academic works).
Rupert
benefits of physcology
Irfan Reply
who define first physcology
Irfan
who define first physcology
Irfan
I think it was either Henry James Or William James are the father of psychology.
James
I haven't read the flash cards and memorized them sense 2015.
James
so iv forgotten some info and some has never left me . I'm very passionate about psychology.
James
I study medical psychology
John
the history of psychology usually traces back to ancient chinese practices. but modern times acknowledge Wilhelm Wundt as the first person to establish a school.of thought in psychology.
Rupert
huh
ANGELICA
is it wrong?
Rupert
please can i know what is the difference about io psychology and human resources managment and which is more in demand ?
Reina Reply
What is psychology
Tayyeba Reply
someone who can observe or understand a person's reactions.
Leanne
It is the scientific study of behavior and mental processes
Enos
Sarah is a 30 year old white female. Her occupation is part time chef. She is recently been divorced and has custody of her three children. She has extreme changes in her mood that began to appear in her 20’s but she never went to see a doctor about them. When family tries to talk to her about her u
Ere Reply
listen very carefully for red flags on both sides
Leanne
what does historical racism mean
Unasumming Reply
Means that racism was present in a group of peoples race in there past and maybe even now. Example is the Native Americans and African Americans also Asian people. Anyone of not caucasian resent. So people who are mixed races also experience rascism.
James
racism meaning I would believe is a negative impact on other separate cultures, causing some serious concerns to the human race.
Leanne
How do counsel someone that is kinda depressed, lonely and stuffs but the person is pushing you away with by saying stuffs you don't like? Your age mate
Ajayi Reply
let them figure themselves out met them come to you and listen more than speak
Jascelyn
don't take anything too personal someone who is depressed has more feelings other than being sad. I have been through it we dont alwaya mean the things we say. just give time and space but be there when they are ready.
Deanna
the person should be first dealt with very patience and understanding.......... person's trust should be won first to treat that kind of state if counsel is the priority..... don't rush small steps will lead to get successful results....... after the patient start trusting you conversate with them.
Amisha
OK..... Thank you very much It makes me look desperate for the person's attention Thanks tho
Ajayi
just let them know even if they aren't ready to tall right now, you will be here for them whenever they are.
Deanna
we need to wait till the stage comes when he/ she is ready at mental level to participate as he is the subject and the core of the part of that particular conscelling session
aaqib
A.Are they saying it in general manner or B.they are saying it cz they know something about you personally If it is A. then remember it's a part of counseling...even if you don't agree with their views..it is theirs.. so don't take it personally
PREETI
if it is B. tell them .. we are here to talk about them not you Also remember people takes time with things.. so it does get difficult at times.. but in counseling it's about them not you. And if it affecting you much .. see why is there .. and try to objective about it OR refer them to someone
PREETI
allow them time to always become comfortable by listening to all sorts of thoughts they all sound scattered but if you ,just let it go until their ready to ask for advice ,my thoughts of counciling would go for a few more sessions of them venting until they've had enough to feel they want geedback
Leanne
what is the trait in the given example?
Maya Reply
what's the given example?
SamieMike
my fat cock
Andrea
o.o u seem like u need to read this txt
SamieMike
learning to understand psychological effects of one's mind thoughts
Leanne
A trait is a distinguishing feature of a person's character, either behavioral or physical.   "Your Fat Cock", would be an example of a physical trait. Andrea, at what age did symptoms of your IED first occur Andrea? I'd say my, "This big dick!", outbursts are Circa 2009. & STILL GOING STRONG!
Ashley
mental state and behavior
Mufakir Reply
Feared ,lethargic, self isolated...
Leanne
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QuizOver Reply

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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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