RD Sharma MCQs: Factorization of Polynomials

# RD Sharma MCQs: Factorization of Polynomials - Notes | Study Mathematics (Maths) Class 9 - Class 9

``` Page 1

Q u e s t i o n : 7 7
Mark the correct alternative in each of the following:
If x - 2 is a factor of x
2
+ 3ax - 2a, then a =
a
2
b
- 2
c
1
d
-1
S o l u t i o n :
As is a factor of
i.e.
Hence, correct option is
d.
Q u e s t i o n : 7 8
If x
3
+ 6x
2
+ 4x + k is exactly divisible by x + 2, then k =
a
-6
Page 2

Q u e s t i o n : 7 7
Mark the correct alternative in each of the following:
If x - 2 is a factor of x
2
+ 3ax - 2a, then a =
a
2
b
- 2
c
1
d
-1
S o l u t i o n :
As is a factor of
i.e.
Hence, correct option is
d.
Q u e s t i o n : 7 8
If x
3
+ 6x
2
+ 4x + k is exactly divisible by x + 2, then k =
a
-6
b
-7
c
-8
d
-10
S o l u t i o n :
As is exactly divisible by
i.e., is a factor of f(x).
Therefore,
Hence, the correct option is
c.
Q u e s t i o n : 7 9
If x - a is a factor of x
3
-3x
2
a + 2a
2
x + b, then the value of b is
a
0
b
2
c
1
d
3
S o l u t i o n :
As is a factor of
i.e.
Thus, b = 0
Thus, the correct option is
a.
Q u e s t i o n : 8 0
If x
140
+ 2x
151
+ k is divisible by x + 1, then the value of k is
a
1
b
-3
c
2
d
-2
S o l u t i o n :
As is divisible by
i.e., is a factor of f(x).
Therefore,
Hence, the correct option is
a.
Q u e s t i o n : 8 1
If x + 2 is a factor of x
2
+ mx + 14, then m =
a
7
b
2
c
9
d
14
S o l u t i o n :
Page 3

Q u e s t i o n : 7 7
Mark the correct alternative in each of the following:
If x - 2 is a factor of x
2
+ 3ax - 2a, then a =
a
2
b
- 2
c
1
d
-1
S o l u t i o n :
As is a factor of
i.e.
Hence, correct option is
d.
Q u e s t i o n : 7 8
If x
3
+ 6x
2
+ 4x + k is exactly divisible by x + 2, then k =
a
-6
b
-7
c
-8
d
-10
S o l u t i o n :
As is exactly divisible by
i.e., is a factor of f(x).
Therefore,
Hence, the correct option is
c.
Q u e s t i o n : 7 9
If x - a is a factor of x
3
-3x
2
a + 2a
2
x + b, then the value of b is
a
0
b
2
c
1
d
3
S o l u t i o n :
As is a factor of
i.e.
Thus, b = 0
Thus, the correct option is
a.
Q u e s t i o n : 8 0
If x
140
+ 2x
151
+ k is divisible by x + 1, then the value of k is
a
1
b
-3
c
2
d
-2
S o l u t i o n :
As is divisible by
i.e., is a factor of f(x).
Therefore,
Hence, the correct option is
a.
Q u e s t i o n : 8 1
If x + 2 is a factor of x
2
+ mx + 14, then m =
a
7
b
2
c
9
d
14
S o l u t i o n :
As  is a factor of
Therefore,
Hence, the correct option is
c.
Q u e s t i o n : 8 2
If x - 3 is a factor of x
2
- ax - 15, then a =
a
-2
b
5
c
-5
d
3
S o l u t i o n :
As is a factor of polynomial
i.e.
Therefore,
Hence, the correct option is
a
Q u e s t i o n : 8 3
If x
51
+ 51 is divided by x + 1, the remainder is
a
0
b
1
c
49
d
50
S o l u t i o n :
As is divided by
The remainder will be
Hence, the correct option is
d.
Q u e s t i o n : 8 4
If x + 1 is a factor of the polynomial 2x
2
+ kx, then k =
a
-2
b
-3
c
4
d
2
S o l u t i o n :
As  is a factor of polynomial Therefore,
So,
Hence, the correct option is
d.
Page 4

Q u e s t i o n : 7 7
Mark the correct alternative in each of the following:
If x - 2 is a factor of x
2
+ 3ax - 2a, then a =
a
2
b
- 2
c
1
d
-1
S o l u t i o n :
As is a factor of
i.e.
Hence, correct option is
d.
Q u e s t i o n : 7 8
If x
3
+ 6x
2
+ 4x + k is exactly divisible by x + 2, then k =
a
-6
b
-7
c
-8
d
-10
S o l u t i o n :
As is exactly divisible by
i.e., is a factor of f(x).
Therefore,
Hence, the correct option is
c.
Q u e s t i o n : 7 9
If x - a is a factor of x
3
-3x
2
a + 2a
2
x + b, then the value of b is
a
0
b
2
c
1
d
3
S o l u t i o n :
As is a factor of
i.e.
Thus, b = 0
Thus, the correct option is
a.
Q u e s t i o n : 8 0
If x
140
+ 2x
151
+ k is divisible by x + 1, then the value of k is
a
1
b
-3
c
2
d
-2
S o l u t i o n :
As is divisible by
i.e., is a factor of f(x).
Therefore,
Hence, the correct option is
a.
Q u e s t i o n : 8 1
If x + 2 is a factor of x
2
+ mx + 14, then m =
a
7
b
2
c
9
d
14
S o l u t i o n :
As  is a factor of
Therefore,
Hence, the correct option is
c.
Q u e s t i o n : 8 2
If x - 3 is a factor of x
2
- ax - 15, then a =
a
-2
b
5
c
-5
d
3
S o l u t i o n :
As is a factor of polynomial
i.e.
Therefore,
Hence, the correct option is
a
Q u e s t i o n : 8 3
If x
51
+ 51 is divided by x + 1, the remainder is
a
0
b
1
c
49
d
50
S o l u t i o n :
As is divided by
The remainder will be
Hence, the correct option is
d.
Q u e s t i o n : 8 4
If x + 1 is a factor of the polynomial 2x
2
+ kx, then k =
a
-2
b
-3
c
4
d
2
S o l u t i o n :
As  is a factor of polynomial Therefore,
So,
Hence, the correct option is
d.
Q u e s t i o n : 8 5
If x + a is a factor of x
4
- a
2
x
2
+ 3x - 6a, then a =
a
0
b
-1
c
1
d
2
S o l u t i o n :
As is a factor of polynomial
Therefore,
Hence, the correct option is
a.
Q u e s t i o n : 8 6
The value of k for which x - 1 is a factor of 4x
3
+ 3x
2
- 4x + k, is
a
3
b
1
c
-2
d
-3
S o l u t i o n :
As is a factor of polynomial f(x) = 4x
3
+ 3x
2
- 4x + k
Therefore,
Hence, the correct option is
d.
Q u e s t i o n : 8 7
If x + 2  and x - 1 are the factors of x
3
+ 10x
2
+ mx + n, then the values of m and n are respectively
a
5 and -3
b
17 and -8
c
7 and -18
d
23 and -19
S o l u t i o n :
It is given and are the factors of the polynomial
i.e.,  and
Now
-8 +40 -2m +n = 0 ? -2m +n = -32 ? 2m -n = 32                         . . . (i)
And
Solving equation
i and
ii we get
Page 5

Q u e s t i o n : 7 7
Mark the correct alternative in each of the following:
If x - 2 is a factor of x
2
+ 3ax - 2a, then a =
a
2
b
- 2
c
1
d
-1
S o l u t i o n :
As is a factor of
i.e.
Hence, correct option is
d.
Q u e s t i o n : 7 8
If x
3
+ 6x
2
+ 4x + k is exactly divisible by x + 2, then k =
a
-6
b
-7
c
-8
d
-10
S o l u t i o n :
As is exactly divisible by
i.e., is a factor of f(x).
Therefore,
Hence, the correct option is
c.
Q u e s t i o n : 7 9
If x - a is a factor of x
3
-3x
2
a + 2a
2
x + b, then the value of b is
a
0
b
2
c
1
d
3
S o l u t i o n :
As is a factor of
i.e.
Thus, b = 0
Thus, the correct option is
a.
Q u e s t i o n : 8 0
If x
140
+ 2x
151
+ k is divisible by x + 1, then the value of k is
a
1
b
-3
c
2
d
-2
S o l u t i o n :
As is divisible by
i.e., is a factor of f(x).
Therefore,
Hence, the correct option is
a.
Q u e s t i o n : 8 1
If x + 2 is a factor of x
2
+ mx + 14, then m =
a
7
b
2
c
9
d
14
S o l u t i o n :
As  is a factor of
Therefore,
Hence, the correct option is
c.
Q u e s t i o n : 8 2
If x - 3 is a factor of x
2
- ax - 15, then a =
a
-2
b
5
c
-5
d
3
S o l u t i o n :
As is a factor of polynomial
i.e.
Therefore,
Hence, the correct option is
a
Q u e s t i o n : 8 3
If x
51
+ 51 is divided by x + 1, the remainder is
a
0
b
1
c
49
d
50
S o l u t i o n :
As is divided by
The remainder will be
Hence, the correct option is
d.
Q u e s t i o n : 8 4
If x + 1 is a factor of the polynomial 2x
2
+ kx, then k =
a
-2
b
-3
c
4
d
2
S o l u t i o n :
As  is a factor of polynomial Therefore,
So,
Hence, the correct option is
d.
Q u e s t i o n : 8 5
If x + a is a factor of x
4
- a
2
x
2
+ 3x - 6a, then a =
a
0
b
-1
c
1
d
2
S o l u t i o n :
As is a factor of polynomial
Therefore,
Hence, the correct option is
a.
Q u e s t i o n : 8 6
The value of k for which x - 1 is a factor of 4x
3
+ 3x
2
- 4x + k, is
a
3
b
1
c
-2
d
-3
S o l u t i o n :
As is a factor of polynomial f(x) = 4x
3
+ 3x
2
- 4x + k
Therefore,
Hence, the correct option is
d.
Q u e s t i o n : 8 7
If x + 2  and x - 1 are the factors of x
3
+ 10x
2
+ mx + n, then the values of m and n are respectively
a
5 and -3
b
17 and -8
c
7 and -18
d
23 and -19
S o l u t i o n :
It is given and are the factors of the polynomial
i.e.,  and
Now
-8 +40 -2m +n = 0 ? -2m +n = -32 ? 2m -n = 32                         . . . (i)
And
Solving equation
i and
ii we get
m = 7 and n = - 18
Hence, the correct option is
c
Q u e s t i o n : 8 8
Let f(x)  be a polynomial such that f -
1
2
= 0, then a factor of f(x) is
a
2x - 1
b
2x + 1
c
x - 1
d
x + 1
S o l u t i o n :
Let f(x) be a polynomial such that
i.e.,  is a factor.
On rearranging  can be written as
Thus,  is a factor of f(x).
Hence, the correct option is
b.
Q u e s t i o n : 8 9
When x
3
- 2x
2
+ ax - b is divided by x
2
- 2x - 3, the remainder is x - 6. The values of a and b are respectively
a
-2, -6
b
2 and -6
c
-2 and 6
d
2 and 6
S o l u t i o n :
If the reminder (x -6) is subtracted from the given polynomial  then rest of part of this polynomial is exactly divisible by x
2
- 2x - 3.
Therefore,
Now,
Therefore,  are factors of polynomial p(x).
Now,
And
and
Solving
i and
ii we get
Hence, the correct option is
c.
Q u e s t i o n : 9 0
One factor of x
4
+ x
2
- 20 is x
2
+ 5. The other factor is
a
( )
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## Mathematics (Maths) Class 9

88 videos|397 docs|109 tests

## Mathematics (Maths) Class 9

88 videos|397 docs|109 tests

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