Class 9 Polynomials Ex 2.1

Question 1: 

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

(i)\(\) 4x² – 3x + 7

Solution

One variable is involved in given polynomial which is \(x\)

Therefore, it is a polynomial in one variable \(x\).

ii)\(\displaystyle y^2 + \sqrt 2\)

Solution

One variable is involved in given polynomial which is \(y\)

Therefore, it is a polynomial in one variable '\(y\)'.

iii)\(\displaystyle 3\sqrt t + t\sqrt 2\)

Solution

No. It can be observed that the exponent of variable \(t\) in term

\(3\sqrt t\) is \(\dfrac{1}{2}\), which is not a whole number.

Therefore, this expression is not a polynomial.

iv)\(\displaystyle y + \dfrac{2}{y}\)

Solution

The power of variable ‘\(y\)’ is \(-1\) which is not a whole number.

Therefore, it is not a polynomial in one variable

No. It can be observed that the exponent of variable \(y\) in term

\(\dfrac{2}{y}\) is \(-1\), which is not a whole number.

Therefore, this expression is not a polynomial.

(v) \(\displaystyle x^{10} + y^3 + t^{50}\)

Solution 

In the given expression there are \(3\) variables which are ‘\(x, y, t\)’ involved.

Therefore, it is not a polynomial in one variable.

Question 2:

Write the coefficients of x² in each of the following:

(i) \(2 + x^2 + x\)

(ii) \(2 - x^2 + x^3\)

(iii)\(\dfrac{\pi}{2} x^2 + x\)

(iv) \(√2x - 1\)

Solution 2:

(i) \(2 + x^2 + x\)

= \(2 + 1(x^2 )+ x\)

The coefficient of \(x^2\) is \(1\).

(ii) \(2 - x^2 + x^3\)

\(2 - 1(x^2) + x^3\)

The coefficient of \(x^2\) is \(-1\).

(iii)\(\dfrac{\pi}{2} x^2 + x\)

The coefficient \(x^2\) is \(\dfrac{\pi}{2}\).

(iv) \(\sqrt 2x - 1  = 0x^2 + \sqrt 2x - 1\)

The coefficient of \(x^2\) is \(0\).

Question 3:

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

Solution 3 :

Binomial of degree \(35\) : \(x^{35}+x^{34}\)

Monomial of degree \(100\) : \(x^{100}\).

Question 4:

Write the degree of each of the following polynomials:

(i) \(5x^3 + 4x^2 + 7x\)

Solution

Degree of this polynomial is \(3\)

(ii) \(4 - y^2\)

Solution

Degree of this polynomial is \(2\).

(iii) \(5t - \sqrt 7\)

Solution

Degree of this polynomial is \(1\).

(iv) \(3\)

Solution 

Degree of a constant polynomial is always \(0\).

Question 5: 

Classify the following as linear, quadratic and cubic polynomial:

(i) \(x^2 + x\)
\(x^2 + x\) is a quadratic polynomial as its highest degree is 2.

(ii) \(x - x^3\)
\(x - x^3\) is a cubic polynomial as its highest degree is 3.

(iii) \(y + y^2 + 4\)
\(y + y^2 + 4\) is a quadratic polynomial as its highest degree is 2.

(iv) 1 + x
1 + x is a linear polynomial as its degree is 1.

(v) 3t
3t is a linear polynomial as its degree is 1.

(vi) \(r^2\)
\(r^2\) is a quadratic polynomial as its degree is 2.

(vii) \(7x^2 + 7x^3\)
 \(7x^2 + 7x^3\) is a cubic polynomial as highest its degree is 3.