Polynomials Test

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Polynomials Test - 41

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Weekly Quiz Competition

Question 1
1 / -0

Which of the following is a cubic polynomial?

A
$$p(x) = y^3 - 27$$

B
$$p(y) = x^3 - 27$$

C
$$p(x) = x^3 - 27$$

D
$$p(y) = 27$$

Solution

$$\textbf{Step-1: Apply the concept of the polynomial.}$$

$$\text{A cubic polynomial is a polynomial of degree}$$ $$3.$$
$$\text{For example,}$$ $$x^3 - 1, 4a^3 - 100a^2 + a - 6$$ $$\text{and}$$ $$m^2n + mn^2$$ $$\text{are all cubic polynomials.}$$

$$\text{(a) In the polynomial}$$ $$p(x),$$ $$\text{the variable is}$$ $$x.$$
$$\text{So, the cubic polynomial should be in terms of}$$ $$x$$ $$\text{only.}$$
$$\text{But here,}$$ $$p(x)=y^3-27,$$ $$\text{in which the cubic polynomial is in terms of variable}$$ $$y$$.

$$\text{Therefore,}$$ $$p(x)=y^3-27$$ $$\text{is not a cubic polynomial.}$$

$$\text{(b) In the polynomial}$$ $$p(y),$$ $$\text{the variable is}$$ $$y$$. $$\text{So, the cubic polynomial should be in terms of}$$ $$y$$
$$\text{ only, But here,}$$ $$p(y)=x^3-27,$$ $$\text{in which the cubic polynomial is in terms of variable}$$ $$x$$.

$$\text{Therefore,}$$ $$p(y)=x^3-27$$ $$\text{is not a cubic polynomial.}$$

$$\text{(c) In the polynomial}$$ $$p(x),$$ $$\text{the variable is}$$ $$x$$. $$\text{So, the cubic polynomial should be in terms of}$$ $$x$$
$$\text{only, But here,}$$ $$p(x)=x^3-27,$$ $$\text{in which the cubic polynomial is in terms of variable}$$ $$x$$.

$$\text{So,}$$ $$p(x)=x^3-27$$ $$\text{is a cubic polynomial.}$$

. $$\text{(d) In the polynomial}$$ $$p(y),$$ $$\text{the variable is}$$ $$y$$. $$\text{So, the cubic polynomial should be in terms of}$$ $$y$$
$$\text{ only, But here,}$$ $$p(y)=27,$$ $$\text{in which there is only constant}$$ $$x$$.

$$\text{Therefore,}$$ $$p(y)=27$$ $$\text{is not a cubic polynomial.}$$

$$\textbf{Hence, correct option is C}$$
Question 2
1 / -0

Which of the following is a cubic polynomial?

A
$$p(x) = x^2 - 16$$

B
$$p(x) = x- 16$$

C
$$p(x) = x^3 - 27$$

D
$$p(x) = 27^3$$

Solution

$$\textbf{Step-1: Apply the concept of the polynomial.}$$

$$\text{The general form of cubic polynomial is}$$ $$p(x)=ax^3+bx^2+cx+d$$ $$\text{where}$$

$$x$$ $$\text{is a variable and}$$ $$a,b,c,d$$ $$\text{are constants.}$$ $$\text{The first term of this polynomial has power}$$ $$3,$$

$$\text{the second term of this polynomial has power 2}$$

$$ \text{and the third term of this polynomial has power 1}$$

$$\text{ and therefore, the degree of the polynomial is the largest exponent that is 3}$$

$$\text{Similarly, in the polynomial}$$ $$p(x)=x^3-27,$$ $$\text{the first term of this polynomial has power 3}$$

$$\text{and the second term of this polynial has power 0}$$

$$\text{ and therefore, the degree of the polynomial is the largest exponent that is 3}$$

$$\text{So, the polynomial}$$ $$p(x)=x^3-27$$ $$\text{is a cubic polynomial.}$$

$$\textbf{Hence, correct option is C}$$
Question 3
1 / -0

Which of the following is NOT a cubic polynomial?

A
$$p(x) = \dfrac{x^3}{3}$$

B
$$p(x) = \dfrac{0x}{x} + 3x^3$$

C
$$p(x) = x^3 - x^2$$

D
$$p(x) = (x-1)(x+1)$$

Solution

$$\textbf{Step-1: Apply the concept of polynomial.}$$

$$\text{The general form of cubic polynomial is}$$ $$p(x)=ax^3+bx^2+cx+d,$$
$$\text{where}$$ $$x$$ $$\text{is a variable and}$$ $$a,b,c,d$$ $$\text{are constants. }$$
$$\text{The first term of this polynomial has power}$$ $$3,$$
$$\text{the second term of this polynomial has power}$$ $$2$$ $$\text{and }$$
$$\text{the third term of this polynomial has power}$$ $$1$$

$$\text{Therefore, the degree of the polynomial is the largest exponent that is}$$ $$3$$.

$$\text{option (D)}$$ $$p(x)=(x-1)(x+1)=x^2-1$$ $$\text{hence it has degree}$$ $$2$$

$$\textbf{Hence, it is NOT a cubic polynomial.}$$
Question 4
1 / -0

Which of the following is a cubic polynomial?

A
$$p(x) = x^3 - 3^3$$

B
$$p(x) = x - 3$$

C
$$p(x) = 3$$

D
$$p(x) = 3^3$$

Solution

$$\textbf{Step-1: Apply the concept of polynomial}$$

$$\text{The general form of cubic polynomial is}$$ $$p(x)=ax^3+bx^2+cx+d$$ $$\text{where}$$ $$x$$ $$\text{is a variable and}$$

$$a,b,c,d$$ $$\text{are constants. The first term of this polynomial}$$ $$\text{has power}$$ $$3,$$

$$\text{the second term of this polynomial has power 2}$$

$$\text{the third term of this polynomial has power 1}$$

$$\text{and therefore, the degree of the polynomial is the largest exponent that is}$$ $$3$$.

$$\text{Similarly, in the polynomial}$$ $$p(x)=x^3-3^3$$

$$\text{the first term of this polynomial has power 3}$$

$$\text{the second term of this polynomial has power 0}$$

$$\text{and therefore, the degree of the polynomial is the largest exponent that is}$$ $$3$$.

$$\text{So, the polynomial}$$ $$p(x)=x^3-3^3$$ $$\text{is a cubic polynomial.}$$

$$\textbf{Hence, the correct option is A}$$
Question 5
1 / -0

Which of the following is NOT a cubic polynomial?

A
$$p(x) = x^2$$

B
$$p(x) = x^3$$

C
$$p(x) = ax^3$$

D
$$p(x) = bx^3$$

Solution

$$\textbf{Step-1: Apply the concept of polynomial.}$$

$$\text{The general form of cubic polynomial is}$$ $$p(x)=ax^3+bx^2+cx+d$$ $$\text{where}$$ $$x$$ $$\text{is a variable and}$$
$$a,b,c,d$$ $$\text{are constants.}$$ $$\text{The first term of this polynomial has power}$$ $$3,$$
$$\text{the second term of this polynomial has power}$$ $$2$$
$$\text{and the third term of this polynomial has power}$$ $$1$$
$$\text{and therefore,}$$ $$\text{the degree of the polynomial is the largest exponent that is}$$ $$3$$.

$$\text{But, in the polynomial}$$ $$p(x)=x^2,$$ $$\text{the degree of the polynomial is 2}$$

$$\text{So, the polynomial}$$ $$p(x)=x^2$$ $$\text{is not a cubic polynomial.}$$

$$\textbf{Hence, correct option is A}$$
Question 6
1 / -0

Zero polynomial can be written as ________.

A
$$p(x) = x$$

B
$$p(x) = 1$$

C
$$p(x) = 0$$

D
$$p(x) = x^2$$

Solution

Consider the polynomial, $$p(x) = ax^2 + bx +c$$ , if $$a=b=c= 0$$ then the expression becomes zero polynomial.

Therefore, the constant polynomial whose coefficients are all equal to $$0$$ is called a zero polynomial.

Hence, zero polynomial can be written as $$p(x) = 0$$.
Question 7
1 / -0

A polynomial whose coefficients are all equal to _______ is called zero polynomial.

A
$$1$$

B
$$2$$

C
$$3$$

D
$$0$$

Solution

Consider the polynomial, $$p(x) = ax^2 + bx +c$$ , if $$a=b=c= 0$$ then the expression becomes zero polynomial.

Therefore, zero polynomial can be written as $$p(x) = 0$$.

Hence, the constant polynomial whose coefficients are all equal to $$0$$ is called a zero polynomial.
Question 8
1 / -0

Which of the following does NOT represent a zero polynomial?

A
$$p(x) = 0$$

B
$$p(x) = 0.x^0$$

C
$$p(x) = x^0$$

D
$$p(x) = x^0 - 1$$

Solution

Consider the polynomial, $$p(x) = ax^2 + bx +c$$ , if $$a=b=c= 0$$ then the expression becomes zero polynomial.
Therefore, the constant polynomial whose coefficients are all equal to $$0$$ is called a zero polynomial.

Zero polynomial can be written as $$p(x) = 0$$.

Now, the polynomial $$p(x)=x^0$$ or $$p(x)=1$$ is a constant polynomial but the coefficient is not equal to $$0$$ and therefore it is not of the form $$p(x) = 0$$.

Hence, the polynomial $$p(x)=x^0$$ is not a zero polynomial
Question 9
1 / -0

The zero polynomial is the _________ identity of the additive group of polynomials.

A
multiplicative

B
additive

C
multiplicative inverse

D
additive inverse

Solution

Zero polynomial has all coefficients $$=0$$
If $$p(x)$$ is a zero polynomial then $$p(x)=0$$
It is the additive identity of additive group of polynomials.

Hence, $$Op-B$$ is correct.
Question 10
1 / -0

$$3x^3-2x-x+3$$ is an example of

A
square polynomial

B
monomial

C
binomial

D
cubic polynomial

Solution

$$3x^3-2x-x+3=3x^3-3x+3$$ is an example of cubic polynomial.
A cubic polynomial is a polynomial of degree $$3$$.
Since the highest degree is $$3$$.

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Polynomials Test - 41

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Polynomials Test - 41

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Question 1

$$p(x) = y^3 - 27$$

$$p(y) = x^3 - 27$$

$$p(x) = x^3 - 27$$

$$p(y) = 27$$

Solution

Question 2

$$p(x) = x^2 - 16$$

$$p(x) = x- 16$$

$$p(x) = x^3 - 27$$

$$p(x) = 27^3$$

Solution

Question 3

$$p(x) = \dfrac{x^3}{3}$$

$$p(x) = \dfrac{0x}{x} + 3x^3$$

$$p(x) = x^3 - x^2$$

$$p(x) = (x-1)(x+1)$$

Solution

Question 4

$$p(x) = x^3 - 3^3$$

$$p(x) = x - 3$$

$$p(x) = 3$$

$$p(x) = 3^3$$

Solution

Question 5

$$p(x) = x^2$$

$$p(x) = x^3$$

$$p(x) = ax^3$$

$$p(x) = bx^3$$

Solution

Question 6

$$p(x) = x$$

$$p(x) = 1$$

$$p(x) = 0$$

$$p(x) = x^2$$

Solution

Question 7

$$1$$

$$2$$

$$3$$

$$0$$

Solution

Question 8

$$p(x) = 0$$

$$p(x) = 0.x^0$$

$$p(x) = x^0$$

$$p(x) = x^0 - 1$$

Solution

Question 9

multiplicative

additive

multiplicative inverse

additive inverse

Solution

Question 10

square polynomial

monomial

binomial

cubic polynomial

Solution

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