Conditions of a Polynomial

Q. Question 1
Which of the following expressions are polynomials? Justify your answer.
I. $8$
II. $\sqrt{3}{x}^2-2x$
III. $1-\sqrt{5x}$
IV. $\frac{1}{5x^{-2}}+5x+7$
V. $\frac{(x-2)(x-4)}{x}$
VI. $\frac{1}{x+1}$
VII. $\frac{1}{7}{a}^3-\frac{2}{\sqrt{3}}{a}^2+4a-7$
VIII. $\frac{1}{2x}$

Q.

If $x^g+\frac{1}{x^a}$ is a polynomial, what could be the value of $a$?

1. only 1

2. any positive integer

3. any negative integer

4. any integer

Q.

Which of the following is not a polynomial?

1. $x+x^2$

2. $x+\sqrt{3}$

3. $\sqrt[3]{3y}+y^2$

4. $x$

Q. In the polynomial $f\left(x\right)=a_0{x}^n+a_1{x}^{n-1}+...+a_{n-1}x+a_n,\mathrm{where}{a}_0,a_1,a_2,...,a_n$ are the real numbers, if $a_0=a_1=a_2=...=a_{n-1}=0\mathrm{and}{a}_n\ne 0,$ then this is

1. Not a polynomial
2. A linear polynomial
   
3. A constant polynomial
4. A zero polynomial

Q.

Which of the following is a true statement ?

(a) $x^2+5x-3$ is a linear polynomial.

(b) $x^2+4x-1$ is a binomial.

(c) x+1 is a monomial.

(d) $5x^3$ is a monomial.

Q.

Question 5
Classify the following as constant, linear, quadratic and cubic polynomials.

(i) $2–x^2+x^3$

(ii) $3x^3$

(iii) $5t-\sqrt{7}$

(iv) $4–5y^2$

(v) $3$

(vi) $2+x$

(vii) $y^3-y$

(viii) $1+x+x^2$

(ix) $t^2$

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

Q.

Which of the following is a polynomial in 1 variable?

1. 4x + 2

2. 4x + 2 = 0

3. x + y

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

Q.

Is every polynomial a rational expression?

Q. Check whether $y+y^3$ and $\sqrt[3]{3y}+y^3$ are polynomials or not?
(2 Marks)

Q. Question 1
Which of the following expressions are polynomials in one variable and which are not?
(i) $4x^2-3x+7$
(ii) $y^2+\sqrt{2}$
(iii) $3\sqrt{t}+t\sqrt{2}$
(iv) $y+\frac{2}{y}$
(v) $x^{10}+y^3+t^{50}$

Q.

Give an example of a polynomial, which is Trinomial of degree $2$

Q.

Which of the following expressions are polynomials in one variable?

(1)$4x^2-3x+7$

(2)$y^2+\sqrt{2}$

(3)$y+\frac{2}{y}$

(4)$x^{10}+y^3+t^{50}$

1. 2 & 3

2. 1, 2 & 3

3. 1 and 2

4. 2 and 3

Q. The graph of a second degree polynomial is in the shape of a ___ .

Q.

Which of the following is a polynomial expression in one variable?

1. 4x+2=0

2. x+y

3. 4x+2

4. 

Q. What are the conditions for an Algebraic expression to be called as polynomial?

Q.

Write the polynomial in index form $(5,0,0,0,-1).$

Q. Which of the following polynomials are binomials?
(1) $p\left(x\right)=x+1$
(2) $q\left(x\right)=x^3+x$
(3) $r\left(x\right)=\sqrt{2}+x+x^2$
(4) $r\left(u\right)=u+u^2-2$

1. 1
2. 3 & 4
3. 1 & 2
4. None of these

Q.

Why is a first degree polynomial in two variables ax + by + c = 0, called a linear equation?

1. It has infinitely many solutions

2. The geometrical representation is a straight line

3. It has two variables

4. None of the above

Q. The expression 2x+5x is a polynomial in two variables.

1. True
2. False

Q. Which among the following is a polynomial in one variable?

1. $2x^{\pi }$
2. $y^{-2}$
3. $4x$
4. $4x^{\frac{3}{2}}$

Q.

Which of the following polynomials are binomials?

(1) p(x) = x + 1

(2) q(x) = $x^3+x$

(3)r(x) = $\sqrt{2}+x+x^2$

(4)r(u) = $u+u^2-2$

1. 1

2. 2

3. 3

4. 1 and 2

Q. ‘0’ is a polynomial.

1. True
2. False

Q. Why should we add the powers of the variables, when two or more variables are used in a polynomial? Is there any specific reason for doing so?

Q.

State True or False:

Every polynomial is a binomial.

1. True
2. False

Q. If $x^a+\frac{1}{x^b}$ is a polynomial and if it is given that a = b, then the value of $a$ is .

1. \N
2. 1
3. -1
4. 2

Q.

Which of the following polynomials are trinomials?

(1) p(x) = $x+x^2+\pi$

(2)q(x)= $\sqrt{2}+x-x^2$

(3) r(x) = $x^3+x^2+1$

(4)t(y) = $y^4+y+5$

1. 1, 2 and 3

2. 1 and 2

3. All of these

4. 1

Q.

Justify if the expression $\sqrt{3}{x}^2-2x$ is polynomial or not.

Q. Which of the following is not a polynomial?

1. $x+x^2$
2. $x$
3. $x+\sqrt{3}$
4. $\sqrt[3]{3y}+y^2$

Q. Which of the following expressions is a polynomial in one variable?

1. $x^{\pi }+x^{-2}$
2. $\pi x^3+3x^2+9$
3. $2x^4+4x^{3/2}+2$
4. $3x^2+4xy+\sqrt{2}$

Q. Which of the following expressions is a polynomial?

1. $\sqrt{x}$
2. $\left(\frac{3}{4}\right)x$
3. $x^{3/4}$
4. $x^{-2}$