RD Sharma 2020 solution class 9 chapter 6 Factorization of polynomial Expressions MCQS

MCQS

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Question 1:

Which one of the following is a polynomial?

(a) x22-2x2

(b) 2x-1

(c) x2+3x3/2x

(4) x-1x+1

Answer 1:

(a) x22-2x2=x22-2x-2

Exponent of x cannot be negative.
Therefore, it is not a polynomial.

(b) 2x-1

Exponent of x must be a whole number.
Therefore, it is not a polynomial.

(c) x2+3x3/2x=x2+3x32-12=x2+3x

It is a polynomial.

(d)  x-1x+1

It is not a polynomial.


Hence, the correct option is (c).





Question 2:

Degree of the polynomial f(x) = 4x4 + 0x3 + 0x5 + 5x + 7 is
(a) 4
(b) 5
(c) 3
(d) 7

Answer 2:

Given: f(x) = 4x4 + 0x+ 0x5 + 5x + 7 = 4x4 + 5x + 7

Degree is the highest power of x in the polynomial.

Here, the highest power of x is 4.

Thus, degree of the polynomial is 4.

Hence, the correct option is (a).
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Question 23:

Let f(x)  be a polynomial such that f-12 = 0, then a factor of f(x) is
(a) 2x − 1

(b) 2x + 1

(c) x − 1

(d) x + 1

Answer 23:

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).


Question 24:

When x3 − 2x2 + ax − b is divided by x2 − 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

Answer 24:

If the reminder (x −6) is subtracted from the given polynomial then rest of part of this polynomial is exactly divisible by x2 − 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).


Question 25:

One factor of x4 + x2 − 20 is x2 + 5. The other factor is

(a) x2 − 4

(b) x − 4

(c) x2 − 5

(d) x + 2

Answer 25:

It is given that is a factor of the polynomial


Here, reminder is zero. Therefore, is a factor of polynomial.
Thus, the correct option is (a).


Question 26:

If (x − 1) is a factor of polynomial f(x) but not of g(x) , then it must be a factor of

(a) f(x) g(x)

(b) −f(x) + g(x)

(c) f(x) − g(x)

(d) f(x)+g(x) g(x)

Answer 26:

Asis a factor of polynomial f(x) but not of g(x)
Therefore
Now,
Let
Now

Therefore (x − 1) is also a factor of f(x).g(x).
Hence, the correct option is (a).


Question 27:

(x+1) is a factor of xn + 1 only if

(i) n is an odd integer

(ii) n is an even integer

(iii) n is a negative integer

(iv) n is a positive integer

Answer 27:

The linear polynomial is a factor of only if
If n is odd integer, then
Hence, the correct option is (a).


Question 28:

If x2 + x + 1 is a factor of the polynomial 3x3 + 8x2 + 8x + 3 + 5k, then the value of k is

(a) 0

(b) 2/5

(c) 5/2

(d) −1

Answer 28:

Let be the given polynomial,
Since is the factor of f(x). Therefore, re`maider will be zero.
Now,

Now,

Hence, the correct option is (b).


Question 29:

If (3x − 1)7  = a7x7 + a6x6 + a5x5 +...+ a1x + a0, then a7 + a5 + ...+a1 + a0 =

(a) 0

(b) 1

(c) 128

(d) 64

Answer 29:

Given that,

Putting
We get

Hence, the correct option is (c).


Question 30:

If both x − 2 and x-12 are factors of px2 + 5x + r, then

(a) p = r

(b) p + r = 0

(c) 2p + r = 0

(d) p + 2r = 0

Answer 30:

As and are the factors of the polynomial
i.e., and
Now,

And 


From equation (i) and (ii), we get

Hence, the correct option is (a).


Question 18:

If x51 + 51 is divided by x + 1, the remainder is

(a) 0

(b) 1

(c) 49

(d) 50

Answer 18:

As is divided by
The remainder will be

Hence, the correct option is (d).


Question 19:

If x + 1 is a factor of the polynomial 2x2 + kx, then k =

(a) −2

(b) −3

(c) 4

(d) 2

Answer 19:

As is a factor of polynomial Therefore,
So,

Hence, the correct option is (d).


Question 20:

If x + a is a factor of x4a2x2 + 3x − 6a, then a =

(a) 0

(b) −1

(c) 1

(d) 2

Answer 20:

Asis a factor of polynomial
Therefore, 


Hence, the correct option is (a).


Question 21:

The value of k for which x − 1 is a factor of 4x3 + 3x2 − 4x + k, is

(a) 3

(b) 1

(c) −2

(d) −3

Answer 21:

As is a factor of polynomial f(x) = 4x3 + 3x2 − 4x + k
Therefore,

Hence, the correct option is (d).


Question 22:

If x + 2  and x − 1 are the factors of x3 + 10x2 + 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

Answer 22:

It is given and are the factors of the polynomial

i.e., and
Now

-8+40-2m+n=0-2m+n=-322m-n=32                         ...(i)
And

Solving equation (i) and (ii) we get
m = 7 and n = − 18
Hence, the correct option is (c)

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Question 31:

If x2 − 1 is a factor of ax4 + bx3 + cx2 + dx + e, then

(a) a + c + e = b + d

(b) a + b +e = c + d

(c) a + b + c = d + e

(d) b + c + d = a + e

Answer 31:

Asis a factor of polynomial

Therefore, 

And 
f(1) = 0
a14+b13+c12+d1+e=0a+b+c+d+e=0
And


Hence,
The correct option is (a).

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