NCERT Solutions for Class 9 Maths Chapter 2 – Polynomials Ex 2.1

NCERT Solutions for Class 9 Maths Chapter 2 – Polynomials Ex 2.1 are given here in simple, step-by-step form. NCERT stands for National Council of Educational Research and Training, the body that makes your textbook. A polynomial is an algebraic expression made of terms, where each term has a variable raised to a whole number power. Exercise 2.1 is the starting point of the chapter. It teaches you to spot a polynomial, and to find its degree, its terms and its coefficients.

This exercise has 5 questions. You will check whether an expression is a polynomial in one variable, write down the coefficient of a chosen term, give examples of polynomials of a certain degree, and state the degree of different polynomials. Every question below is solved in short steps, with the final answer in bold. Students can use this page for homework and revision, parents can use it to check the work at home, and teachers can use it in class. A free printable PDF of these solutions is also available for offline study.

NCERT Solutions for Class 9 Maths Chapter 2 – Polynomials Ex 2.1

Exercise 2.1 – Questions and Answers

Q1. Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.
(i) 4x2 − 3x + 7   (ii) y2 + √2   (iii) 3√t + t√2   (iv) y + 2/y   (v) x10 + y3 + t50

Answer:

(i) 4x2 − 3x + 7
There is only one variable x, and all its powers (2, 1, 0) are whole numbers.
It is a polynomial in one variable.

(ii) y2 + √2
There is only one variable y, and √2 is just a constant number.
It is a polynomial in one variable.

(iii) 3√t + t√2
The term 3√t is 3t1/2, and the power 1/2 is not a whole number.
It is not a polynomial.

(iv) y + 2/y
The term 2/y is 2y−1, and the power −1 is not a whole number.
It is not a polynomial.

(v) x10 + y3 + t50
The powers are all whole numbers, but there are three different variables x, y and t.
It is a polynomial, but not a polynomial in one variable.

Q2. Write the coefficients of x2 in each of the following.
(i) 2 + x2 + x   (ii) 2 − x2 + x3   (iii) (π/2)x2 + x   (iv) √2x − 1

Answer: The coefficient of x2 is the number multiplying x2.

(i) 2 + x2 + x. Here x2 means 1 × x2, so the coefficient is 1.
(ii) 2 − x2 + x3. Here −x2 means −1 × x2, so the coefficient is −1.
(iii) (π/2)x2 + x. The coefficient of x2 is π/2.
(iv) √2x − 1. There is no x2 term, so the coefficient of x2 is 0.

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

Answer:
A binomial has two terms, and degree 35 means the highest power is 35.
Example of a binomial of degree 35: 3x35 − 5 (any two-term example with highest power 35 is correct).

A monomial has one term, and degree 100 means the power is 100.
Example of a monomial of degree 100: 7x100 (any single-term example with power 100 is correct).

Q4. Write the degree of each of the following polynomials.
(i) 5x3 + 4x2 + 7x   (ii) 4 − y2   (iii) 5t − √7   (iv) 3

Answer: The degree is the highest power of the variable.

(i) 5x3 + 4x2 + 7x. The highest power is 3, so the degree is 3.
(ii) 4 − y2. The highest power is 2, so the degree is 2.
(iii) 5t − √7. The highest power of t is 1, so the degree is 1.
(iv) 3. This is a constant. A non-zero constant has degree 0.

Q5. Classify the following as linear, quadratic and cubic polynomials.
(i) x2 + x   (ii) x − x3   (iii) y + y2 + 4   (iv) 1 + x   (v) 3t   (vi) r2   (vii) 7x3

Answer: Linear means degree 1, quadratic means degree 2, and cubic means degree 3.

(i) x2 + x. Highest power 2. Quadratic.
(ii) x − x3. Highest power 3. Cubic.
(iii) y + y2 + 4. Highest power 2. Quadratic.
(iv) 1 + x. Highest power 1. Linear.
(v) 3t. Highest power 1. Linear.
(vi) r2. Highest power 2. Quadratic.
(vii) 7x3. Highest power 3. Cubic.

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

i 4x2−3x+7ii y2+2iii 3t+t2iv y+2yvx10+y3+t50

Ans

(i)   4x2−3x+7→It is a polynomial in variable x.(ii)  y2+2→ It is a polynomial in variable y.(iii) 3t+t2=3t12+t2It is not a polynomial in variable t because power of t is in fraction i.e., 12.(iv) y+2y=y+2y−1It is not a polynomial in variable y because power of y is negative i.e., −1.(v)x10+y3+t50→It is not a polynomial in one variable because there are three variables i.e., x, y and t.

Q.2

Write the coefficients of x2 in each of the following:i 2+x2+xii 2−x2+x3iii π2x2+xiv 2x−1

Ans

(i) 2+x2+xCoefficient of x2 is 1.(ii) 2−x2+x3Coefficient of x2 is  −1.(iii) π2x2+xCoefficient of x2 is π2.(iv) 2x−1Coefficient of x2 is 0.

Q.3 Give one example each of a binomial of degree 35, and of a monomial of degree 100.

Ans Binomial of degree 35 is x35 + 15. Monomial of degree 100 is x100.

Q.4

Write the degree of each of the following polynomials:i 5x3+4x2+7xii 4−y2 iii 5t−7iv 3

Ans

(i) 5x3+4x2+7x,The degree of the given polynomial is 3 becuase the heighest power of x is 3.(ii) 4−y2, The degree of the given polynomial is 2 becuase heighest power of y is 2.(iii) 5t−7, The degree of the given polynomial is 1 becuase heighest power of t is 1.(iv) 3The degree of the given polynomial is 0 becuase of constant term.

Q.5

Classify the following as linear, quadratic and cubicpolynomials:(i) x2+x    (ii)x−x3(iii) y+y2+4 (iv) 1+x(v) 3t (vi) r2 (vii) 7x3

Ans

(i) x2+xThis polynomial is quadratic polynomial because  degree of the polynomial is 2.(ii) x−x3This polynomial is cubic polynomial because  degree of the polynomial is 3.(iii) y+y2+4This polynomial is quadratic polynomial because  degree of the polynomial is 2.(iv) 1+xThis polynomial is linear polynomial because  degree of the polynomial is 1.(v) 3tThis polynomial is linear polynomial because  degree of the polynomial is 1.(vi) r2This polynomial is quadratic polynomial because  degree of the polynomial is 2.(vii) 7x3This polynomial is cubic polynomial because  its degree is 3.

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