# Division – Exercise 3.4.3 – Class IX

1. Arrange the following in descending powers of x:

(i) x2 + x5+ 2x3 + x2 – 2 + 3x

(ii) x8 + x9 + x12 – 3x7 + x2 + 1

(iii) x2y6 + x5y2 + x6y4 + 3x2y8

Solution:

(i) x2 + x5+ 2x3 + x2 – 2 + 3x

x5 + 2x3 + x2 + 3x-2

(ii) x8 + x9 + x12 – 3x7 + x2 + 1

x12 + x9 + x8 – 3x7 + x +1

(iii) x2y6 + x5y2 + x6y4 + 3x2y8

x6y4 + x5y2 + x2y6 + 3x2y8

1. Perform the following divisions:

(i) 3x2 + 4x –  4 by x+2

x+2 ) 3x2 + 4x – 4 (3x – 2

3x2 + 6x

(-)      (-)

———————–

-2x – 4

-2x – 4

(+)   (+)

————————

0

(ii)  2x2 – 9x + 9 by x – 3

x – 3 ) 2x2 – 9x + 9 (2x – 3

2x2 – 6x

(-)      (+)

—————————————

-3x + 9

-3x + 9

(+)   (-)

———————————

0

(iii) 2x2 + 2x + 11 by x + 3

x + 3 ) 2x2 + 2x + 11 (2x – 4

2x2 + 6x

(-)     (-)

————————————–

– 4x + 11

– 4x + 12

(+)  (-)

——————————-

-1

(iv) 3x2 – 13x + 11 by x – 3

x – 3) 3x2 – 13x + 11 (3x – 4

3x2 – 9x

(-)     (+)

————————————–

-4x + 11

-4x + 12

(+)   (-)

——————————-

-1

(v) x3 – 5x2 + 8x – 4 by x – 2

x – 2 ) x3 – 5x2 + 8x – 4 (x2 – 3x + 2

x3 – 2x2

(-)   (+)

———————————————–

-3x2 + 8x

-3x2 + 6x

(+)   (-)

——————–

2x – 4

2x – 4

(-)  (+)

——————–

0

(vi) 6x3 + x– 23x + 12 by 2x – 3

2x – 3 ) 6x3 + x2 – 23x + 12  (3x2 + 5x – 4

6x3 – 9x2

(-)    (+)

—————————————

10x2 –  23x

10x2 – 15x

(-)   (+)

——————–

-8x + 12

-8x + 12

(+)  (-)

——————–

0

(vii) 2x2 – 7x + 6 by x – 2

x – 2 ) 2x2 – 7x + 6 (2x – 3

2x2 – 4x

(-)    (+)

————————

-3x + 4

-3x + 4

(+)   (-)

————————-

0

(viii) x4 – 0.x3 – 3x2 + 0.x – 4 by x + 2

x+2 ) x4 – 0.x3 – 3x2 + 0.x – 4 (x3 – 2x2 + x – 2

x4 + 2x3

(-)   (-)

———————-

-2x3 – 3x2

-2x3 – 4x2

(+)   (+)

———————-

x2 + 0.x

x2 + 2x

(-)  (-)

—————————

-2x – 4

-2x – 4

(+)  (+)

—————————

0

(ix) 3x2 + x3 + 4 by x + 2

x + 2) x3 + 3x2 +0.x + 4 (x2 + x – 2

x3 + 2x2

(-)    (-)

—————————

x2 + 0.x

x2 + 2x

(-)   (-)

—————————-

-2x + 4

-2x  – 4

(+)   (+)

—————————-

0

(xi) x4 – 16 by x – 2

x – 2 ) x4 + 0.x3 + 0.x2 + 0.x – 16 (x3 + 2x2 + 4x + 8

x4 – 2x3

(-)   (+)

———————–

2x3 + 0.x2

2x3 – 4x2

(-)   (+)

————————

4x2 + 0.x

4x2 – 8x

(-)   (+)

———————–

8x – 16

8x – 16

(-)  (+)

——————-

0

(xii) x3 – 1 by x – 1

x – 1)x3 + 0.x2 + 0.x  – 1 (x2 + x + 1

x3 – x2

(-)    (+)

————————

x2 + 0.x

x2 – x

(-)     (+)

————————-

x – 1

x – 1

(-) (+)

————————

0

(xiii) 8x3 – 27 by 2x – 3

2x – 3 ) 8x3 – 0.x2 – 0.x – 27 ( 4x2 + 6x + 9

8x3 – 12x2

(-)      (+)

—————————

12x2 – 0.x

12x2 – 18x

(-)        (+)

—————————

18x  – 27

18x – 27

(-)     (+)

—————————

0

3 . Divide

(i)x5 + a5 by x + a

x + a ) x5 + 0.x4.a + 0.x3.a2 + 0.x2.a3 + 0.x.a4 + a5 (x4 – x3a + x2a2 – xa3 + a4

x5 + x4.a

(-)   (-)

———————————–

-x4.a + 0.x3.a2

-x4.a2 – x3a2

(+)       (+)

—————————————

x3a2 + 0.x2.a3

x3a2 + x2.a3

(-)     (-)

————————————-

-x2 a3 + 0.x.a4

-x2.a3 – xa4

(+)      (+)

———————————

xa4 + a5

xa4 + a5

(-)     (-)

——————————

0

(ii) x7 – y7 by x – y

(iiii) x9 + y9 by x3 + y3

x3 + y3 ) x9 + 0.x6.y3 + 0.x3.y6 + y9 (x6 – x3y3 + y6

x9  + x6y3

(-)    (-)

——————————

-x6y3 + 0.x3y6

-x6y3 – x3y6

(+)     (+)

—————————

x3y6 + y9

x3y6 + y9

(-)     (-)

—————————–

0

1. Divide a + b by a1/3 + b1/3

a1/3 + b1/3 ) a + 0.a2/3.b1/3 + 0.a1/3.b2/3 + b (a2/3 – a1/3 + b2/3

a + a2/3b1/3

(-)    (-)

—————————–

– a2/3b1/3 + 0. a1/3b2/3

– a2/3b1/3 – a1/3b2/3

(+)        (+)

—————————–

a1/3b2/3 + b

a1/3b2/3 + b

(-)          (-)

—————————–

0

1. Divide a2– b2 by a1/2 – b1/2

Solution:

a1/2 – b1/2 ) a2 +0. a3/2 b1/2 + 0.ab +0.a1/2.b3/2 – b2(a3/2 + ab1/2 + a1/2.b3/2 + b2

a2 – a3/2b1/2

(-)  (+)

—————————-

+ a3/2 b1/2 + 0.ab

+ a3/2 b1/2 – ab

(-)           (+)

—————————

+ab – 0.a1/2.b3/2

+ab – a1/2.b3/2

(-)      (+)

—————————-

+ a1/2.b3/2 – b2

+ a1/2.b3/2 – b2

(-)           (+)

—————————-

0

1. Which of the following are visible by x+a? (that is the division leaves 0 remainder?

(i)x3-a3

x+a ) x3 + 0.x2.a + 0.xa2 – a3(x2 – xa + a2

x3 + x2a

(-)  (-)

——————————–

-x2a + 0.xa2

-x2a – xa2

(+)   (+)

—————————–

+ xa2 – a3

+xa2 + a3

(-)   (-)

—————————–

-2a3

Therefore (x+a) is not divisible by x3 – a3 since it leaves a remainder = -2a3

(ii) x4 – a4

x+a ) x4 + 0.x3a + 0.x2a2 + 0.xa3 + a4(x3 – x2a +xa2

x4 + x3a

(-)   (-)

————————–

-x3a + 0.x2a2

-x3a  – x2.a2

(+)  (+)

———————–

+x2a2 + 0.xa3

+x2a2 + xa3

(-)      (-)

——————–

-xa3+a4

-xa3+a4

(+)  (-)

———————-

0

Therefore (x+a) is divisible by x4 – a4

(iii) x7 + a7

Therefore, (x+a) divides x7 + a7

(iv) x8 + a8

Thus, (x + a) divides x8 + a8