NCERT Solutions Class 9 Maths 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+
   (v) x¹⁰+y³+t⁵⁰
   Sol:- (i) 4x²–3x+7 is a polynomial in one variable. The equation 4x²–3x+7 can be written as 4x²–3x¹+7x⁰ and since x is the only variable in the given equation and the powers of x (i.e., 2, 1 and 0) are whole numbers.
   (ii) y²+√2 is a polynomial in one variable. The equation y²+√2 can be written as y²+√2y⁰ Since y is the only variable in the given equation and the powers of y (i.e., 2 and 0) are whole numbers.
   (iii) 3√t+t√2 is not a polynomial in one variable. The equation 3√t+t√2 can be written as 3t^1/2+√2t though, t is the only variable in the given equation, the powers of t (i.e.,1/2) is not a whole number.
   (iv) y+ is not a polynomial in one variable. The equation y+2/y an be written as y+2y^-1 though, y is the only variable in the given equation, the powers of y (i.e.,-1) is not a whole number.
   (v) x¹⁰+y³+t⁵⁰  is not a polynomial in one variable. Here, in the equation x¹⁰+y³+t⁵⁰ though, the powers, 10, 3, 50, are whole numbers, there are 3 variables used in the expression x¹⁰+y³+t⁵⁰.
   
2. Write the coefficients of x² in each of the following:
   (i) 2+x²+x
   (ii) 2-x²+x³
   (iii) x²+x
   (iv) √2x-1
   Sol:- (i) The coefficient of x² in 2+x²+x is 1
   (ii) The coefficient of x² in 2–x²+x³ is -1.
   (iii) The coefficient of x² in x²+x is .
   (iv) The coefficient of x² in √2x-1 is 0.
   
3. Give one example each of a binomial of degree 35, and of a monomial of degree 100.
   Sol:- Binomial of degree 35: A polynomial having two terms and the highest degree 35 is called a binomial of degree 35. Eg.,  3x³⁵+5.
   Monomial of degree 100: A polynomial having one term and the highest degree 100 is called a monomial of degree 100. Eg.,  4x¹⁰⁰.
   
4. Write the degree of each of the following polynomials:
   (i) 5x³+4x²+7x
   (ii) 4 – y²
   (iii) 5t –√7
   (iv) 3
   Ans:- (i) The degree of 5x³+4x²+7x is 3 as 3 is the highest power of x in the equation.
   (ii) The degree of 4–y² is 2 as 2 is the highest power of y in the equation.
   (iii) The degree of 5t–√7 is 1 as 1 is the highest power of y in the equation.
   (iv) The degree of 3 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
   Sol:- (i) The highest power of x²+x is 2
   So, x²+x is a quadratic polynomial.
   (ii) The highest power of x–x³ is 3
   So, x–x³ is a cubic polynomial.
   (iii) The highest power of y+y²+4 is 2
   So,  y+y²+4is a quadratic polynomial.
   (iv) The highest power of 1+x is 1
   So, 1+x is a linear polynomial.
   (v) The highest power of 3t is 1
   So, 3t is a linear polynomial.
   (vi) The highest power of r² is 2
   So,  r²is a quadratic polynomial.
   (vii) The highest power of 7x³ is 3
   So,  7x³ is a cubic polynomial.