NCERT Solutions Class 9 Maths Chapter 2 Polynomials Ex 2.1

NCERT Solutions Class 9 Maths
Chapter – 2 (Polynomials)

The NCERT Solutions in English Language for Class 9 Mathematics Chapter – 2 Polynomials Exercise 2.1 has been provided here to help the students in solving the questions from this exercise.

Chapter 2: Polynomials

Exercise – 2.1

1. 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 + $\mathbf{\frac{2}{y}}$
(v) x¹⁰+ y³+ t⁵⁰

Answer –
(i) 4x²– 3x + 7
We have 4x² – 3x + 7 = 4x² – 3x + 7x⁰
It is a polynomial in one variable i.e., x, because each exponent of x is a whole number.

(ii) y²+√2
We have y² + √2 = y² + √2y⁰
It is a polynomial in one variable i.e., y, because each exponent of y is a whole number.

(iii) 3√t + t√2
We have 3√t + t√2 = 3 √t^1/2 + √2.t
It is not a polynomial, because one of the exponents of t is $\frac{1}{2}$ , which is not a whole number.

(iv) y + $\mathbf{\frac{2}{y}}$
We have y + $\frac{2}{y}$ = y + 2.y^-1It is not a polynomial, because one of the exponents of y is -1, which is not a whole number.

(v) x¹⁰+ y³+ t⁵⁰We have x¹⁰+ y³+ t⁵⁰Here, exponent of every variable is a whole number, but x¹⁰ + y³ + t⁵⁰ is a polynomial in x, y and t, i.e., in three variables. So, it is not a polynomial in one variable.

2. Write the coefficients of x² in each of the following.
(i) 2 + x²+ x
(ii) 2 – x² + x³
(iii) $\mathbf{\frac{\pi}{2}}$ x² + x
(iv) √2 x – 1

Answer –
(i) 2 + x²+ x
The equation 2 + x²+ x can be written as 2 + (1)x²+ x.
The coefficient of x² is 1.

(ii) 2–x²+x³The equation 2 – x²+ x³can be written as 2 + (–1)x²+ x³.
The coefficient of x² is -1.

(iii) $\mathbf{\frac{\pi}{2}}$ x²+ x
The equation $\frac{\pi }{2}$ x²+ x.
The coefficient of x² is $\frac{\pi }{2}$ .

(iii) √2x – 1
The equation √2x – 1 can be written as 0x²+ √2x – 1 [Since 0x² is 0]
The coefficient of x² is 0.

3. Give one example each of a binomial of degree 35 and of a monomial of degree 100.
Answer – Binomial of degree 35 – A polynomial having two terms and the highest degree of 35 is called a binomial of degree 35. E.g., 3x³⁵+ 5
Monomial of degree 100 – A polynomial having one term and the highest degree of 100 is called a monomial of degree 100. E.g., 4x¹⁰⁰

4. Write the degree of each of the following polynomials.
(i) 5x³+ 4x²+ 7x
(ii) 4 – y²
(iii) 5t – √7
(iv) 3
Answer –
(i) 5x³+ 4x²+ 7x
The given polynomial is 5x³ + 4x² + 7x.
The highest power of the variable x is 3.
So, the degree of the polynomial is 3.

(ii) 4 – y²The given polynomial is 4- y². The highest
power of the variable y is 2.
So, the degree of the polynomial is 2.

(iii) 5t – √7
The given polynomial is 5t – √7 . The highest power of variable t is 1. So, the degree of the polynomial is 1.

(iv) 3
Since, 3 = 3x° [∵ x° = 1]
So, the degree of the polynomial is 0.

5. 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 polynomial – A polynomial of degree one is called a linear polynomial.
Quadratic polynomial – A polynomial of degree two is called a quadratic polynomial.
Cubic polynomial – A polynomial of degree three is called a cubic polynomial.

(i) x²+ x
The highest power of x²+ x is 2
The degree is 2.
Hence, x²+ x is a quadratic polynomial.

(ii) x – x³The highest power of x – x³is 3.
The degree is 3.
Hence, x – x³ is a cubic polynomial.

(iii) y + y²+ 4
The highest power of y + y²+ 4 is 2.
the degree is 2
Hence, y + y²+ 4 is a quadratic polynomial

(iv) 1 + x
The highest power of 1 + x is 1.
The degree is 1.
Hence, 1 + x is a linear polynomial.

(v) 3t
The highest power of 3t is 1.
The degree is 1.
Hence, 3t is a linear polynomial.

(vi) r²The highest power of r²is 2.
The degree is 2.
Hence, r²is a quadratic polynomial.

(vii) 7x³The highest power of 7x³is 3.
The degree is 3.
Hence, 7x³ is a cubic polynomial.

NCERT Solutions Class 9 Maths Chapter 2 Polynomials Ex 2.1