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Exercise - 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,    (ii) y² + √2,    (iii) 3√t + t√2,    (iv) y + 2/y,    (v) x¹⁰ + y³ + t⁵⁰.

Solution :-
(i) 4x²- 3x + 7 
   Yes, this polynomial is in the form of only one variable of x.   

(ii) y² + √2, 
   Yes, this polynomial is in the form of only one variable of y.    
        
(iii) 3√t + t√2,
   No, this polynomial is not in the form of exact form 
   because √t which is not a whole number.   
        
(iv) y + 2/y,
   No, this polynomial not in the form of exact form 
   because 2/y which is not a whole number.     
        
(v) x¹⁰ + y³ + t⁵⁰.    
   No, this polynomial not in the form of exact form 
   because there x, y and t are three variables.
    

Question-2 :-  Write the coefficients of x² in each of the following:
(i) 2 + x² + x,    (ii) 2 - x² + x³,    (iii) πx²/2 + x,    (iv) √2x - 1.

Solution :-
(i) 2 + x² + x 
   The coefficient of x² is 1. 
        
(ii) 2 - x² + x³
   The coefficient of x² is -1.    
        
(iii) πx²/2 + x
   The coefficient of x² is π/2. 
        
(iv) √2x - 1.
   The coefficient of x² is 0.
    

Question-3 :-  Give one example each of a binomial of degree 35, and of a monomial of degree 100.

Solution :-
   The binomial degree of 35 = x³⁵ + 2
   The binomial degree of 100 = 3y + y¹⁰⁰ - 24   
    

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

Solution :-
(i) 5x³ + 4x² + 7x
   The highest power of the polynomial is 3. So, degree is 3.

(ii) 4 - y² 
   The highest power of the polynomial is 2. So, degree is 2. 
        
(iii) 5t - √7
   The highest power of the polynomial is 1. So, degree is 1.
        
(iv) 3
   There is no any variable. So, degree is 0.  
    

Question-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.

Solution :-
(i) x² + x
   The highest degree of the polynomial is 2. 
   So, this polynomial is quadratic polynomial.   
         
(ii) x - x³
   The highest degree of the polynomial is 3. 
   So, this polynomial is cubic polynomial.

(iii) y + y² + 4 
   The highest degree of the polynomial is 2. 
   So, this polynomial is quadratic polynomial.

(iv) 1 + x
   The highest degree of the polynomial is 1. 
   So, this polynomial is linear polynomial.

(v) 3t
   The highest degree of the polynomial is 1. 
   So, this polynomial is linear polynomial.

(vi) r²
   The highest degree of the polynomial is 2. 
   So, this polynomial is quadratic polynomial.

(vii) 7x³
   The highest degree of the polynomial is 3. 
   So, this polynomial is cubic polynomial.
    
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