- 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
- 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 + x2 – 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
- 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
- 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
- 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
1 thought on “Division – Exercise 3.4.3 – Class IX”
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