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This mock test of Test: Polynomials - 1 for Class 9 helps you for every Class 9 entrance exam.
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QUESTION: 1

A cubic polynomial is a polynomial with degree

Solution:

A cubic polynomial is a polynomial of degree 3. A univariate cubic polynomial has the form f(x) = a_{3} x^{3 }+ a_{2} x^{2} + a_{1} x + a_{0}. An equation involving a cubic polynomial is called a cubic equation. A closed-form solution known as the cubic formula exists for the solutions of an arbitrary cubic equation.

QUESTION: 2

A polynomial of degree 5 in x has at most

Solution:

A polynomial of degree 5 is of the form *p*(*x*) =, where *a, b, c, d, e, *and *f* are real numbers and *a* ≠ 0.

Thus, *p*(*x*) can have at most 6 terms and at least one term containing.

QUESTION: 3

The coefficient of x^{3} in the polynomial 5 + 2x + 3x^{2} – 7x^{3} is

Solution:

QUESTION: 4

The quadratic polynomial whose sum of zeroes is 3 and the product of zeroes is –2 is :

Solution:

Sum of zeros = 3/1

-b/a = 3/1 .....................(1)

Product of zeros = -2/1

c/a = -2/1 ...................(2)

From equation (1) and (2)

a = 1

-b = 3, b = -3

c = -2

The required quadratic equation is

ax^2+by+c

= x^{2}-3x-2

QUESTION: 5

A linear polynomial :-

Solution:

A number of zeroes of an n -degree polynomial = n.

First, a linear polynomial is in the form of ax + b, a≠0, a,b ∈R

The degree of the polynomial = highest degree of the terms

So here the highest degree is 1.

Hence, Linear polynomial has only one zero.

QUESTION: 6

If x + y = 3, x^{2} + y^{2} = 5 then xy is

Solution:

QUESTION: 7

When the polynomial x^{3} + 3x^{2} + 3x + 1 is divided by x + 1, the remainder is :-

Solution:

The zero of x + 1 is –1

And by remainder theorem, when

p(x) = x^{3} + 3x^{2} + 3x + 1 is divided by x + 1, then remainder is p(–1).

∴ p(–1) = (–1)^{3} + 3 (–1)^{2} + 3(–1) + 1

= –1 + (3 × 1) + (–3) + 1

= –1 + 3 – 3 + 1

= 0

Thus, the required = 0

QUESTION: 8

If the polynomial 2x^{3} – 3x^{2} + 2x – 4 is divided by x – 2, then the remainder is :-

Solution:

QUESTION: 9

The value of k for which x – 1 is a factor of the polynomial 4x^{3}+ 3x^{2} – 4x + k is :-

Solution:

X - 1 is a factor of 4x^{3} + 3x^{2} -4x +k

then x=1 is one root of 4x^{3} + 3x^{2} -4x +k

put x= 1

4x^{3} +3x^{2} -4x +k = 0

=> 4 (1)^{3} +3 (1)^{2}-4 (1) +k =0

=> 4 + 3 - 4 + k = 0

=> k = -3

QUESTION: 10

The value of k for which x + 1 is a factor of the polynomial x^{3} + x^{2} + x + k is :-

Solution:

QUESTION: 11

The value of m for which x – 2 is a factor of the polynomial x^{4} – x^{3} + 2x^{2} – mx + 4 is :-

Solution:

QUESTION: 12

The factors of 2x^{2} – 3x – 2 are :-

Solution:

2x^{2} – 3x – 2

2x^{2} - 4x + x - 2 = 0

2x^{2}(x-2) +1(x-2) = 0

(2x+1) (x-2)

QUESTION: 13

If x + 2 is a factor of x^{3} – 2ax^{2} + 16, then value of a is

Solution:

QUESTION: 14

The factors of x^{3} – 2x^{2} – 13x – 10 are :-

Solution:

QUESTION: 15

The expanded form of (2x – 3y – z)^{2} is :-

Solution:

QUESTION: 16

The expanded form of (x + y + 2z)^{2} is :-

Solution:

QUESTION: 17

The expanded form of (x+1/3)^{3} is :-

Solution:

QUESTION: 18

x^{3} + y^{3} + z^{3} – 3xyz is :-

Solution:

We know that x^{3} + y^{3} + z^{3} – 3xyz = (x + y + z) (x^{2} + y^{2} + z^{2} – xy – yz – zx).

If x + y + z = 0, then x^{3} + y^{3} + z^{3} – 3xyz = 0 or x^{3} + y^{3} + z^{3} = 3xyz.

QUESTION: 19

(a – b)^{3} + (b – c)^{3} + (c – a)^{3} is equal to :-

Solution:

Let x = (a – b), y = (b – c) and z = (c – a)

Consider, x + y + z = (a – b) + (b – c) + (c – a) = 0

⇒ x^{3} + y^{3} + z^{3} = 3xyz

That is (a – b)^{3} + (b – c)^{3} + (c – a)^{3} = 3(a – b)(b – c)(c – a)

QUESTION: 20

is equal to :-

Solution:

QUESTION: 21

√2 is a polynomial of degree

Solution:

The highest power of the variable is known as the degree of the polynomial.

√2x^0 = √2

hence the degree of the polynomial is zero.

QUESTION: 22

The degree of the polynomial 4x^{4}+0x^{3}+0x^{5}+5x+74x^{4}+0x^{3}+0x^{5}+5x+7 is

Solution:

The degree of the polynomial 4x^{4}+0x^{3}+0x^{5}+5x+74x^{4}+0x^{3}+0x^{5}+5x+7 is

QUESTION: 23

The degree of the zero polynomial is

Solution:

The degree of zero polynomial is not defined, because, in zero polynomial, the coefficient of any variable is zero i.e., Ox^{2 }or Ox^{5}, etc. Hence, we cannot exactly determine the degree of the variable.

QUESTION: 24

The value of the polynomial 5x−4x^{2}+3, when x = −1 is

Solution:

Let p (x) = 5x – 4x2 + 3 …(i)

On putting x = -1 in Eq. (i), we get

p(-1) = 5(-1) -4(-1)2 + 3= - 5 - 4 + 3 = -6

QUESTION: 25

If p(x) = x + 3, then p(x) + p(-x) is equal to

Solution:

p(x)=x+3

p(-x)=-x+3

p(x)+p(-x)=x+3-x+3=6

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