Class 9 Exam > Class 9 Notes > Extra Documents & Tests for Class 9 > Factorization of Polynomials - Exercise 6.3
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Page 1
Q u e s t i o n : 1 6
In each of the following, using the remainder theorem, find the remainder when fx
is divided by gx
and verify the result by actual division: 1 -8
1. f(x) = x
3
+ 4x
2
- 3x + 10, g(x) = x + 4
S o l u t i o n :
Let us denote the given polynomials as
We have to find the remainder when is divided by .
By the remainder theorem, when is divided by the remainder is
Now we will show by actual division
So the remainder by actual division is 22
Q u e s t i o n : 1 7
f(x) = 4x
4
- 3x
3
- 2x
2
+ x - 7, g(x) = x - 1
S o l u t i o n :
Let us denote the given polynomials as
We have to find the remainder when is divided by .
By the remainder theorem, when is divided by the remainder is
Now we will show remainder by actual division
So the remainder by actual division is -7
Page 2
Q u e s t i o n : 1 6
In each of the following, using the remainder theorem, find the remainder when fx
is divided by gx
and verify the result by actual division: 1 -8
1. f(x) = x
3
+ 4x
2
- 3x + 10, g(x) = x + 4
S o l u t i o n :
Let us denote the given polynomials as
We have to find the remainder when is divided by .
By the remainder theorem, when is divided by the remainder is
Now we will show by actual division
So the remainder by actual division is 22
Q u e s t i o n : 1 7
f(x) = 4x
4
- 3x
3
- 2x
2
+ x - 7, g(x) = x - 1
S o l u t i o n :
Let us denote the given polynomials as
We have to find the remainder when is divided by .
By the remainder theorem, when is divided by the remainder is
Now we will show remainder by actual division
So the remainder by actual division is -7
Q u e s t i o n : 1 8
f(x) = 2x
4
- 6x
3
+ 2x
2
- x + 2, g(x) = x + 2
S o l u t i o n :
Let us denote the given polynomials as
We have to find the remainder when is divided by .
By the remainder theorem, when is divided by the remainder is
Now we will calculate the remainder by actual division
So the remainder by actual division is 92
Q u e s t i o n : 1 9
f(x) = 4x
3
- 12x
2
+ 14x - 3, g(x) 2x - 1
S o l u t i o n :
Let us denote the given polynomials as
We have to find the remainder when is divided by .
By the remainder theorem, when is divided by the remainder is
Now we will calculate the remainder by actual division
So the remainder by actual division is
Q u e s t i o n : 2 0
f(x) = x
3
- 6x
2
+ 2x - 4, g(x) = 1 - 2x
S o l u t i o n :
Let us denote the given polynomials as
We have to find the remainder when is divided by .
By the remainder theorem, when is divided by the remainder is
Page 3
Q u e s t i o n : 1 6
In each of the following, using the remainder theorem, find the remainder when fx
is divided by gx
and verify the result by actual division: 1 -8
1. f(x) = x
3
+ 4x
2
- 3x + 10, g(x) = x + 4
S o l u t i o n :
Let us denote the given polynomials as
We have to find the remainder when is divided by .
By the remainder theorem, when is divided by the remainder is
Now we will show by actual division
So the remainder by actual division is 22
Q u e s t i o n : 1 7
f(x) = 4x
4
- 3x
3
- 2x
2
+ x - 7, g(x) = x - 1
S o l u t i o n :
Let us denote the given polynomials as
We have to find the remainder when is divided by .
By the remainder theorem, when is divided by the remainder is
Now we will show remainder by actual division
So the remainder by actual division is -7
Q u e s t i o n : 1 8
f(x) = 2x
4
- 6x
3
+ 2x
2
- x + 2, g(x) = x + 2
S o l u t i o n :
Let us denote the given polynomials as
We have to find the remainder when is divided by .
By the remainder theorem, when is divided by the remainder is
Now we will calculate the remainder by actual division
So the remainder by actual division is 92
Q u e s t i o n : 1 9
f(x) = 4x
3
- 12x
2
+ 14x - 3, g(x) 2x - 1
S o l u t i o n :
Let us denote the given polynomials as
We have to find the remainder when is divided by .
By the remainder theorem, when is divided by the remainder is
Now we will calculate the remainder by actual division
So the remainder by actual division is
Q u e s t i o n : 2 0
f(x) = x
3
- 6x
2
+ 2x - 4, g(x) = 1 - 2x
S o l u t i o n :
Let us denote the given polynomials as
We have to find the remainder when is divided by .
By the remainder theorem, when is divided by the remainder is
Now we will calculate remainder by actual division
So the remainder is
Q u e s t i o n : 2 1
f(x) = x
4
- 3x
2
+ 4, g(x) = x - 2
S o l u t i o n :
Let us denote the given polynomials as
We have to find the remainder when is divided by .
By the remainder theorem, when is divided by the remainder is
We will calculate remainder by actual division
So the remainder is 8
Q u e s t i o n : 2 2
f(x) = 9x
3
- 3x
2
+ x - 5, g(x) = x -
2
3
S o l u t i o n :
Let us denote the given polynomials as
We have to find the remainder when is divided by .
By the remainder theorem, when is divided by the remainder is
Remainder by actual division
Page 4
Q u e s t i o n : 1 6
In each of the following, using the remainder theorem, find the remainder when fx
is divided by gx
and verify the result by actual division: 1 -8
1. f(x) = x
3
+ 4x
2
- 3x + 10, g(x) = x + 4
S o l u t i o n :
Let us denote the given polynomials as
We have to find the remainder when is divided by .
By the remainder theorem, when is divided by the remainder is
Now we will show by actual division
So the remainder by actual division is 22
Q u e s t i o n : 1 7
f(x) = 4x
4
- 3x
3
- 2x
2
+ x - 7, g(x) = x - 1
S o l u t i o n :
Let us denote the given polynomials as
We have to find the remainder when is divided by .
By the remainder theorem, when is divided by the remainder is
Now we will show remainder by actual division
So the remainder by actual division is -7
Q u e s t i o n : 1 8
f(x) = 2x
4
- 6x
3
+ 2x
2
- x + 2, g(x) = x + 2
S o l u t i o n :
Let us denote the given polynomials as
We have to find the remainder when is divided by .
By the remainder theorem, when is divided by the remainder is
Now we will calculate the remainder by actual division
So the remainder by actual division is 92
Q u e s t i o n : 1 9
f(x) = 4x
3
- 12x
2
+ 14x - 3, g(x) 2x - 1
S o l u t i o n :
Let us denote the given polynomials as
We have to find the remainder when is divided by .
By the remainder theorem, when is divided by the remainder is
Now we will calculate the remainder by actual division
So the remainder by actual division is
Q u e s t i o n : 2 0
f(x) = x
3
- 6x
2
+ 2x - 4, g(x) = 1 - 2x
S o l u t i o n :
Let us denote the given polynomials as
We have to find the remainder when is divided by .
By the remainder theorem, when is divided by the remainder is
Now we will calculate remainder by actual division
So the remainder is
Q u e s t i o n : 2 1
f(x) = x
4
- 3x
2
+ 4, g(x) = x - 2
S o l u t i o n :
Let us denote the given polynomials as
We have to find the remainder when is divided by .
By the remainder theorem, when is divided by the remainder is
We will calculate remainder by actual division
So the remainder is 8
Q u e s t i o n : 2 2
f(x) = 9x
3
- 3x
2
+ x - 5, g(x) = x -
2
3
S o l u t i o n :
Let us denote the given polynomials as
We have to find the remainder when is divided by .
By the remainder theorem, when is divided by the remainder is
Remainder by actual division
Remainder is -3
Q u e s t i o n : 2 3
f x = 3x
4
+ 2x
3
-
x
2
3
-
x
9
+
2
27
, g x = x +
2
3
S o l u t i o n :
Let us denote the given polynomials as
We have to find the remainder when is divided by .
By the remainder theorem, when is divided by the remainder is
Remainder by actual division
Remainder is 0
Q u e s t i o n : 2 4
If the polynomials 2x
3
+ ax
2
+ 3x - 5 and x
3
+ x
2
- 4x +a leave the same remainder when divided by x -2, find the value of a.
S o l u t i o n :
Let us denote the given polynomials as
Now, we will find the remainders and when and respectively are divided by .
By the remainder theorem, when is divided by the remainder is
By the remainder theorem, when is divided by the remainder is
By the given condition, the two remainders are same. Then we have,
( ) ( )
Page 5
Q u e s t i o n : 1 6
In each of the following, using the remainder theorem, find the remainder when fx
is divided by gx
and verify the result by actual division: 1 -8
1. f(x) = x
3
+ 4x
2
- 3x + 10, g(x) = x + 4
S o l u t i o n :
Let us denote the given polynomials as
We have to find the remainder when is divided by .
By the remainder theorem, when is divided by the remainder is
Now we will show by actual division
So the remainder by actual division is 22
Q u e s t i o n : 1 7
f(x) = 4x
4
- 3x
3
- 2x
2
+ x - 7, g(x) = x - 1
S o l u t i o n :
Let us denote the given polynomials as
We have to find the remainder when is divided by .
By the remainder theorem, when is divided by the remainder is
Now we will show remainder by actual division
So the remainder by actual division is -7
Q u e s t i o n : 1 8
f(x) = 2x
4
- 6x
3
+ 2x
2
- x + 2, g(x) = x + 2
S o l u t i o n :
Let us denote the given polynomials as
We have to find the remainder when is divided by .
By the remainder theorem, when is divided by the remainder is
Now we will calculate the remainder by actual division
So the remainder by actual division is 92
Q u e s t i o n : 1 9
f(x) = 4x
3
- 12x
2
+ 14x - 3, g(x) 2x - 1
S o l u t i o n :
Let us denote the given polynomials as
We have to find the remainder when is divided by .
By the remainder theorem, when is divided by the remainder is
Now we will calculate the remainder by actual division
So the remainder by actual division is
Q u e s t i o n : 2 0
f(x) = x
3
- 6x
2
+ 2x - 4, g(x) = 1 - 2x
S o l u t i o n :
Let us denote the given polynomials as
We have to find the remainder when is divided by .
By the remainder theorem, when is divided by the remainder is
Now we will calculate remainder by actual division
So the remainder is
Q u e s t i o n : 2 1
f(x) = x
4
- 3x
2
+ 4, g(x) = x - 2
S o l u t i o n :
Let us denote the given polynomials as
We have to find the remainder when is divided by .
By the remainder theorem, when is divided by the remainder is
We will calculate remainder by actual division
So the remainder is 8
Q u e s t i o n : 2 2
f(x) = 9x
3
- 3x
2
+ x - 5, g(x) = x -
2
3
S o l u t i o n :
Let us denote the given polynomials as
We have to find the remainder when is divided by .
By the remainder theorem, when is divided by the remainder is
Remainder by actual division
Remainder is -3
Q u e s t i o n : 2 3
f x = 3x
4
+ 2x
3
-
x
2
3
-
x
9
+
2
27
, g x = x +
2
3
S o l u t i o n :
Let us denote the given polynomials as
We have to find the remainder when is divided by .
By the remainder theorem, when is divided by the remainder is
Remainder by actual division
Remainder is 0
Q u e s t i o n : 2 4
If the polynomials 2x
3
+ ax
2
+ 3x - 5 and x
3
+ x
2
- 4x +a leave the same remainder when divided by x -2, find the value of a.
S o l u t i o n :
Let us denote the given polynomials as
Now, we will find the remainders and when and respectively are divided by .
By the remainder theorem, when is divided by the remainder is
By the remainder theorem, when is divided by the remainder is
By the given condition, the two remainders are same. Then we have,
( ) ( )
Q u e s t i o n : 2 5
If the polynomials ax
3
+ 3x
2
- 13 and 2x
3
- 5x + a, when divided by (x - 2) leave the same remainder, find the value of a.
S o l u t i o n :
Let us denote the given polynomials as
Now, we will find the remainders and when and respectively are divided by .
By the remainder theorem, when is divided by the remainder is
By the remainder theorem, when is divided by the remainder is
By the given condition, the two remainders are same. Then we have, R
1
= R
2
Q u e s t i o n : 2 6
Find the remainder when x
3
+ 3x
2
+ 3x + 1 is divided by
i
x + 1
ii
x -
1
2
iii
x
iv
x + p
v
5 + 2x
S o l u t i o n :
Let us denote the given polynomials as
i We will find the remainder when is divided by .
By the remainder theorem, when is divided by the remainder is
ii We will find the remainder when is divided by .
By the remainder theorem, when is divided by the remainder is
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Factorization of Polynomials - Exercise 6.3 Notes
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