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

Exercise 2.1

1. Which of the following expressions are polynomials in one variable and which are not? State the reasons for your answer.

1. \[4{{x}^{2}}-3x+7\]

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

3. \[3\sqrt{t}+t\sqrt{2}\]

4. \[y+\frac{2}{y}\]

5. \[y+2{{y}^{-1}}\]

Ans:

A polynomial in one variable refers to an expression where the exponent of the variable is a whole number.

1. \[4{{x}^{2}}-3x+7\]

In this polynomial, only one variable is involved which is ‘$x$ ’ and the exponents of the variable are all whole numbers.

Therefore, the given expression is a polynomial in one variable ‘$x$ ’.

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

In this polynomial, only one variable is involved which is ‘$y$ ’ and the exponent of the variable is a whole number.

Therefore, the given expression is a polynomial in one variable ‘$y$ ’.

3. \[3\sqrt{t}+t\sqrt{2}\]

In this expression, it is given that the exponent of variable $t$ in term ‘$3\sqrt{t}$ ’ is $\frac{1}{2}$. This exponent is not a whole number. Therefore, the given algebraic expression is not a polynomial in one variable.

4. \[y+\frac{2}{y}\]

We can rewrite this expression as: $y+2{{y}^{-1}}$.

In this expression, it is given that the exponent of the variable $y$ in term ‘$2{{y}^{-1}}$ ’ is $-1$. This exponent is not a whole number. Therefore, the given algebraic expression is not a polynomial in one variable.

5. ${{x}^{10}}+{{y}^{3}}+{{t}^{50}}$ 

In this polynomial, there are $3$ variables involved which are‘$x,y,t$’.

Therefore, the given algebraic expression is not a polynomial in one variable.

2. Write the coefficients of ${{x}^{2}}$in each of the following:

1. $2+{{x}^{2}}+x$ 

2. $2-{{x}^{2}}+{{x}^{3}}$ 

3. $\frac{\pi }{2}{{x}^{2}}+x$ 

4. $\sqrt{2}x-1$ 

Ans: A coefficient is an integer that is multiplied by the variable of a only one term or the terms of a polynomial.

1. $2+{{x}^{2}}+x$

We can rewrite this expression as: $2+1({{x}^{2}})+x$.

Hence, the coefficient of ${{x}^{2}}$ is $1$.

2. $2-{{x}^{2}}+{{x}^{3}}$

We can rewrite this expression as: $2-1({{x}^{2}})+{{x}^{3}}$.

Hence, the coefficient of ${{x}^{2}}$ is $-1$.

3. $\frac{\pi }{2}{{x}^{2}}+x$

In the given expression, the coefficient of ${{x}^{2}}$ is $\frac{\pi }{2}$ .

4. $\sqrt{2}x-1=0{{x}^{2}}+\sqrt{2}x-1$ 

In the given expression, the coefficient of ${{x}^{2}}$ is $0$.

3. Give one example each of a binomial of degree \[35\], and of a monomial of degree $100$ .

Ans: A binomial of degree $35$ refers to a polynomial with two terms and one of the terms has a highest degree of $35$.

Example: ${{x}^{35}}+{{x}^{34}}$

A monomial of degree $100$ refers to a polynomial with only one term and it has a highest degree of $100$.

Example: ${{x}^{100}}$ 

4. Write the degree of each of the following polynomials:

(I) $5{{x}^{3}}+4{{x}^{2}}+7x$ 

(II) $4-{{y}^{2}}$ 

(III) $5t-\sqrt{7}$ 

(IV) $3$ 

Ans: The degree of a polynomial refers to the highest power of a variable in the polynomial.

(i) $5{{x}^{3}}+4{{x}^{2}}+7x$

Here, the highest power of the given variable ‘$x$ ’ is $3$. Hence, the degree of this polynomial is  $3$ .

(ii) $4-{{y}^{2}}$

Here, the highest power of the given variable ‘$y$ ’ is  $2$ . Hence, the degree of this polynomial is  $2$ .

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

Here, the highest power of the given variable ‘$t$ ’ is  $1$ . Hence, the degree of this polynomial is  $1$ .

(iv) $3$

Here, $3$is a constant polynomial. We know the degree of a constant polynomial is always  $0$ . Hence, the degree of this polynomial is  $0$ .

5. Classify the following as linear, quadratic and cubic polynomial:

${{x}^{2}}+x$ 

(i)  $x-{{x}^{3}}$ 

(ii) $y+{{y}^{2}}+4$ 

(iii) $1+x$

(iv) $3t$ 

(v) ${{r}^{2}}$ 

(iv) $7{{x}^{3}}$ 

Ans: The highest exponential power of the variable in a polynomial equation is known as the degree of a polynomial. 

A linear polynomial is a polynomial whose degree is ‘$1$ ’.

A quadratic polynomial is a polynomial whose degree is ‘$2$ ’.

A cubic polynomial is a polynomial whose degree is ‘$3$ ’.

(i) ${{x}^{2}}+x$  

The given expression has a variable $x$ and its degree is $2$.

Hence, it is a quadratic polynomial.

(ii) $x-{{x}^{3}}$   

The given expression has a variable $x$ and its degree is $3$.

Hence, it is a cubic polynomial.

(iii) $y+{{y}^{2}}+4$

The given expression has a variable $y$ and its degree is $2$.

Hence, it is a quadratic polynomial.

(iv) $1+x$ $y+{{y}^{2}}+4$

The given expression has a variable $x$ and its degree is $1$.

Hence, it is a linear polynomial.

(v) $3t$ 

The given expression has a variable $t$ and its degree is $1$.

Hence, it is a linear polynomial.

(vi) ${{r}^{2}}$ 

The given expression has a variable $r$ and its degree is $2$.

Hence, it is a quadratic polynomial.

(vii) $7{{x}^{3}}$

The given expression has a variable $x$ and its degree is $3$.

Hence, it is a cubic polynomial.

Conclusion

Class 9 maths 2.1 Exercise in Chapter 2 aims to strengthen your grasp of polynomial factorization through extensive practice. Mastering these techniques lays a solid groundwork in algebra, which is advantageous for advanced mathematics and numerous practical applications. Class 9 maths exercise 2.1 reinforces the idea that any polynomial can be factored into a product of its factors, an essential skill for solving polynomial equations and simplifying complex expressions.

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