Detailed solution to all exercise questions of NCERT book for CBSE Class 9 Math in Chapter 2 - Polynomials - Exercise 2.1. The exercise comprises 5 questions.

Key concepts covered in this exercise include polynomials in one variable, coefficients of terms in a polynomial, classification of polynomials as monomial, binomial, and classification of polynomials as linear, quadratic, and cubic polynomials.

Question 1

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

(i) 4x^{2}- 3x + 1

(ii) y^{2}+ √2

(iii) 3√t + t√2

(iv) y + \\frac{2}{y})

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

##### Solution

(i) 4x

^{2}- 3x + 1

It is a polynomial in**one**variable.

There is only one variable x and all the exponents are whole numbers.(ii) y

^{2}+ √2

It is a polynomial in**one**variable.

There is only one variable y and its power is a whole number.(iii) 3√t + t√2

It is**NOT**a polynomial in**one**variable.

The exponent of ‘t’ in √t is \\frac{1}{2}) which is not a whole number.(iv) y + \\frac{2}{y})

It is**NOT**a polynomial in**one**variable.

The exponent of y in \\frac{2}{y}) is -1, which is not a whole number.(v) x

^{10}+ y^{3}+ t^{50}

It is**NOT**a polynomial in**one**variable.

There are multiple variables involved i.e., x, y, and t

Question 2

Write the coefficients of x

^{2}in each of the following :

(i) 2 + x^{2}+ x

(ii) 2 - x^{2}+ x^{3}

(iii) \\frac{π}{2})x^{2}+ x

(iv) √2x - 1

##### Solution

(i) 2 + x

^{2}+ x

The coefficient of x^{2}is**1**.(ii) 2 - x

^{2}+ x^{3}

The coefficient of x^{2}is**-1**.(iii) \\frac{π}{2})x

^{2}+ x

The coefficient of x^{2}is**\\frac{π}{2})**.(iv) √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.

:- P(x) = x**Binomial of degree 35**^{35}+ x^{34}:- Q(y) = y**Monomial of degree 100**^{100}

Question 4

Write the degree of each of the following polynomials:

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

(ii) 4 – y^{2}

(iii) 5t - √7

(iv) 3

##### Solution

(i) 5x

^{3}+ 4x^{2}+ 7x

The highest power of x in the polynomial is 3.

The degree of the polynomial is**3**.(ii) 4 – y

^{2}

The highest power of y in the polynomial is 2.

The degree of the polynomial is**2**.(iii) 5t - √7

The highest power of t in the polynomial is 1.

The degree of the polynomial is**1**.(iv) 3

There is no variable in the polynomial.

The degree of the polynomial is**0**.

Question 5

Classify the following as linear, quadratic and cubic polynomials:

(i) x^{2}+ x

(ii) x – x^{3}

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

(iv) 1 + x

(v) 3t

(vi) r^{2}

(vii) 7x^{3}

##### Solution

(i) x

^{2}+ x

The highest power of x in the polynomial is 2.

Therefore the following polynomial is**quadratic**.(ii) x – x

^{3}

The highest power of x in the polynomial is 3.

Therefore the following polynomial is**cubic**.(iii) y + y

^{2}+ 4

The highest power of y in the polynomial is 2.

Therefore the following polynomial is**quadratic**.(iv) 1 + x

The highest power of x in the polynomial is 1.

Therefore the following polynomial is**linear**.(v) 3t

The highest power of t in the polynomial is 1.

Therefore the following polynomial is**linear**.(vi) r

^{2}

The highest power of r in the polynomial is 2.

Therefore the following polynomial is**quadratic**.(vii) 7x

^{3}

The highest power of x in the polynomial is 3.

Therefore the following polynomial is**cubic**.

Class 9 Maths

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