Division of Algebraic Expressions - Exercise 8.5 | Mathematics (Maths) Class 8 PDF Download

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Question:43
Divide the first polynomial by the second in each of the following. Also, write the quotient and remainder:
i
3x
2
 + 4x + 5, x - 2
ii
10x
2
 - 7x + 8, 5x - 3
iii
5y
3
 - 6y
2
 + 6y - 1, 5y - 1
iv
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Question:43
Divide the first polynomial by the second in each of the following. Also, write the quotient and remainder:
i
3x
2
 + 4x + 5, x - 2
ii
10x
2
 - 7x + 8, 5x - 3
iii
5y
3
 - 6y
2
 + 6y - 1, 5y - 1
iv
x
4
 - x
3
 + 5x, x - 1
v
y
4
 + y
2
, y
2
 - 2
Solution:
i 
3x
2
+4x+5
x-2
=
3x(x-2)+10(x-2)+25
(x-2)
=
(x-2)(3x+10)+25
(x-2)
= 3x +10 +
25
(x-2)
Therefore, quotient = 3x +10 and remainder = 25. ii 
10x
2
-7x+8
5x-3
=
2x 5x-3 -
1
5
5x-3 +
47
5
(5x-3)
=
5x-3 2x-
1
5
+
47
5
(5x-3)
= 2x -
1
5
+
47
5
5x-3
Therefore, quotie
(iv)
 x
4
-x
3
+5x
x-1
=
x
3
(x-1)+5(x-1)+5
x-1
=
(x-1)(x
3
+5)+5
x-1
=(x
3
+5)+
5
x-1
Therefore, quotient = x
3
+5 and remainder = 5.
v
 y
4
+y
2
y
2
-2
=
y
2
y
2
-2 +3 y
2
-2 +6
y
2
-2
=
y
2
-2 y
2
+3 +6
y
2
-2
= y
2
+3 +
6
y
2
-2
Therefore, quotient = y
2
+3 and remainder = 6.
Question:44
Find whether the first polynomial is a factor of the second.
i
x + 1, 2x
2
 + 5x + 4
ii
y - 2, 3y
3
 + 5y
2
 + 5y + 2
iii
4x
2
 - 5, 4x
4
 + 7x
2
 + 15
iv
4 - z, 3z
2
 - 13z + 4
v
2a - 3, 10a
2
 - 9a - 5
vi
4y + 1, 8y
2
 - 2y + 1
Solution:
i 
2x
2
+5x+4
x+1
=
2x(x+1)+3(x+1)+1
x+1
=
(x+1)(2x+3)+1
(x+1)
= 2x +3 +
1
x+1
? Remainder = 1Therefore, x +1 is not a  factor of 2x
2
+5x +4
ii) 
3y
3
+5y
2
+5y+2
y-2
=
3y
2
(y-2)+11y(y-2)+27(y-2)+56
y-2
=
(y-2)(3y
2
+11y+27)+56
y-2
=(3y
2
+11y+27)+
56
y-2
? Remainder = 56 ? (y-2) is not a factor of 3y
3
+5y
2
+5y+2.
iii  
4x
4
+
2
+15
4x
2
-5
= 
x
2
(4x
2
-5)+3(4x
2
-5)+30
4x
2
-5
= 
4x
2
-5)(x
2
+3 +30
4x
2
-5
=(x
2
+3 +
30
4x
2
-5
? Remainder = 30Therefore, 4x
2
-5 is not a factor of 4x
4
+7x
2
+15
(iv) 
3z
2
-13z+4
4-z
=
3z
2
-12z-z+4
4-z
=
3z(z-4)-1(z-4)
4-z
=
(z-4)(3z-1)
4-z
=
(4-z)(1-3z)
4-z
=1-3z ? Remainder = 0 ? (4-z) is a factor of 3z
2
-13z+4.
V 
10a
2
-9a-5
2a-3
=
5a(2a-3)+3(2a-3)+4
2a-3
=
(2a-3)(5a+3)+4
2a-3
=(5a+3)+
4
2a-3
? Remainder = 4 ? ( 2a-3) is not a factor of 10a
2
-9a-5.
(vi) 
8y
2
-2y+1
4y+1
=
2y(4y+1)-1(4y+1)+2
4y+1
=
(4y+1)(2y-1)+2
4y+1
= (2y-1)+
2
4y+1
? Remainder = 2 ? 4y+1) is not a factor of 8y
2
-2y+1.
        
 
      
 
      
  
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