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) $\frac{x^2}{2}-\frac{2}{x^2}$

(b) $\sqrt{2x}-1$

(c) $x^2+\frac{3x^{3/2}}{\sqrt{x}}$

(4) $\frac{x-1}{x+1}$

Answer 1:

(a) $\frac{x^2}{2}-\frac{2}{x^2}=\frac{x^2}{2}-2x^{-2}$

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

(b) $\sqrt{2x}-1$

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

(c) $x^2+\frac{3x^{3/2}}{\sqrt{x}}=x^2+3x^{\frac{3}{2}-\frac{1}{2}}=x^2+3x$

It is a polynomial.

(d)  $\frac{x-1}{x+1}$

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 + 0x³ + 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 $f\left(-\frac{1}{2}\right)$ = 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) $\left\{f\left(x\right)+g\left(x\right)\right\} g\left(x\right)$

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 $x-\frac{1}{2}$ 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 x⁴ − a²x² + 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

$$\begin{gathered} -8+40-2m+n=0 \\ \Rightarrow -2m+n=-32 \\ \Rightarrow 2m-n=32 ...\left(i\right) \end{gathered}$$
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
$$\begin{gathered} a{\left(1\right)}^4+b{\left(1\right)}^3+c{\left(1\right)}^2+d\left(1\right)+e=0 \\ \Rightarrow a+b+c+d+e=0 \end{gathered}$$
And


Hence,
The correct option is (a).