NCERT Solutions for Class 9 Maths Chapter 2 Exercise 2.1 - Polynomials

Q1 (i) Is the following expression polynomial in one variable? State reasons for your answer. $4x^2 - 3x + 7$

Answer:

Yes, given the polynomial $4x^2 - 3x + 7$ has only one variable, which is x that is why it is a polynomial in one variable.

Q1 (ii) Is the following expression polynomial in one variable? State reasons for your answer. $y^2 + \sqrt2$

Answer:

Yes, the given polynomial has only one variable, which is y, that is why it is a polynomial in one variable.

Q1 (iii) Is the following expression polynomial in one variable? State reasons for your answer. $3\sqrt t + t\sqrt2$

Answer:

No, because we can observe that the exponent of variable t in the term $3\sqrt t$ is $\frac{1}{2}$, which is not a whole number.
Therefore, the expression given is not a polynomial.

Q1 (iv) Is the following expression polynomial in one variable? State reasons for your answer. $y + \frac{2}{y}$

Answer:

No, because we can observe that the exponent of variable y in the term $\frac{2}{y}$ is $-1$, which is not a whole number. Therefore, the given expression is not a polynomial.

Q1 (v) Is the following expression polynomial in one variable? State reasons for your answer. $x^{10} + y^3 + t^{50}$

Answer:

No, because in the given polynomial $x^{10} + y^3 + t^{50}$ there are 3 variables which are x, y, t. Therefore, the polynomial is not in one variable but in three variables.

Q2 (i) Write the coefficients of $x^2$ in the following: $2 + x^2 +x$

Answer:

Coefficient of $x^2$ in polynomial $2 + x^2 +x$ is 1.

Q2 (ii) Write the coefficients of $x^2$ in the following: $2 - x^2 + x^3$

Answer:

Coefficient of $x^2$ in polynomial $2 - x^2 + x^3$ is -1.

Q2 (iii) Write the coefficients of $x^2$ in the following: $\frac{\pi}{2}x^2 + x$

Answer:

Coefficient of $x^2$ in polynomial $\frac{\pi}{2}x^2 + x$ is $\frac{\pi}{2}$.

Q2 (iv) Write the coefficients of $x^2$ in the following: $\sqrt2 x - 1$

Answer:

Coefficient of $x^2$ in polynomial $\sqrt2 x - 1$ is 0.

Q3 Give one example each of a binomial of degree 35, and of a monomial of degree 100.

Answer:

The degree of a polynomial is the highest power of its variable.

In binomial, there are two terms: Therefore, a binomial of degree 35 is Eg: $x^{35}+1$

In a monomial, there is only one term. Therefore, a monomial of degree 100 can be written as $y^{100}$.

Q4 (i) Write the degree the following polynomial: $5x^3 + 4x^2 + 7x$

Answer:

The degree of a polynomial is the highest power of its variable. Therefore, the degree of the polynomial $5x^3 + 4x^2 + 7x$ is 3.

Q4 (ii) Write the degree the following polynomial: $4 - y^2$

Answer:

The degree of a polynomial is the highest power of its variable. Therefore, the degree of the polynomial $4 - y^2$ is 2.

Q4 (iii) Write the degree the following polynomial: $5t - \sqrt7$

Answer:

The degree of a polynomial is the highest power of its variable. Therefore, the degree of the polynomial $5t - \sqrt7$ is 1.

Q4 (iv) Write the degree the following polynomial: 3

Answer:

The degree of a polynomial is the highest power of its variable. In this case, only a constant value 3 is there, and the degree of a constant polynomial is always 0.

Q5 (i) Classify the following as linear, quadratic and cubic polynomial: $x^2+x$

Answer:

Linear, quadratic, and cubic polynomials have degrees of 1, 2, and 3, respectively. The given polynomial is $x^2+x$ with degree 2. Therefore, it is a quadratic polynomial.

Q5 (ii) Classify the following as linear, quadratic and cubic polynomial: $x - x^3$

Answer:

Linear, quadratic, and cubic polynomials have degrees of 1, 2, and 3, respectively. The given polynomial is $x - x^3$ with degree 3. Therefore, it is a cubic polynomial.

Q5 (iii) Classify the following as linear, quadratic and cubic polynomial: $y + y^2 + 4$

Answer:

Linear, quadratic, and cubic polynomials have degrees of 1, 2, and 3, respectively. The given polynomial is $y + y^2 + 4$ with degree 2. Therefore, it is a quadratic polynomial.

Q5 (iv) Classify the following as linear, quadratic and cubic polynomial: $1 +x$

Answer:

Linear, quadratic, and cubic polynomials have degrees of 1, 2, and 3, respectively. Given polynomial is $1 +x$ with degree 1. Therefore, it is a linear polynomial.

Q5 (v) Classify the following as linear, quadratic and cubic polynomial: $3t$

Answer:

Linear, quadratic, and cubic polynomials have degrees of 1, 2, and 3, respectively. The given polynomial is $3t$ with degree 1. Therefore, it is a linear polynomial.

Q5 (vi) Classify the following as linear, quadratic and cubic polynomial: $r^2$

Answer:

Linear, quadratic, and cubic polynomials have degrees of 1, 2, and 3, respectively. The given polynomial is $r^2$ with degree 2. Therefore, it is a quadratic polynomial.

Q5 (vii) Classify the following as linear, quadratic and cubic polynomial: $7x^3$

Answer:

Linear, quadratic, and cubic polynomials have degrees of 1, 2, and 3, respectively. The given polynomial is $7x^3$ with degree 3. Therefore, it is a cubic polynomial.