Chapter 2. POLYNOMIALS 

Chapter 2 : Polynomials

Exercise 2.1 Complete Solution  Exercise 2.2 Complete Solution

Important note : 1. A polynomial of degree one is called a linear polynomial.The general form of the linear polynomial is   , where  and  are real numbers . For example:  , ………. etc. Note : The expression  etc. are not polynomials . 2. A polynomial of degree two is called a quadratic polynomial. The general form of the quadratic polynomial is   , where and  are real number . For example : ……….etc. 3. A polynomial of degree three is called a cubic polynomial.  The general form of the cubic polynomial is    , where and  are real numbers. For example: ……… etc.

Geometrical Meaning of the Zeroes of a Polynomial : Given a polynomial  of degree , the graph of  intersects the -axis at most  points. Therefore, a polynomial  of degree  has at most  zeroes. (i) A linear polynomial has only one zero . Because, the graph intersects the -axis at one point only. (ii) A quadratic polynomial has  two zeroes . Because, the graph intersects the -axis at two points. (iii) A cubic polynomial has three zeroes . Because, the graph intersects the -axis at three points .

Relationship between Zeroes and Coefficients of a Polynomial : 1. For linear polynomial : If  is the zero of  , then     The zero of the linear polynomial  is =    . 2. For quadratic polynomial : If  and  are the zeroes of the quadratic polynomial  , then  The sum of zeroes = The product of zeroes 3. For cubic polynomial :  If  and  are the zeroes of the quadratic polynomial  , then The sum of zeroes The sum of the product of zeroes taken two at a time    The product of zeroes

Class 10 Maths Chapter 2. POLYNOMIALS

Exercise 2.1

1. The graph of   are given in Fig. 2.10 below, for some polynomials  . Find the number of zeroes of  , in each case .

 

                 (i)

Solution :  (i) The number of zeroes is 0 . Because, the graph does not intersect at the -axis .

            

                 (ii)        

Solution : (ii)  The number of zeroes is 1 as the graph intersects the -axis at one point only .                                                              

   

                   (iii)   

Solution : (iii) The number of zeroes is 3 as the graph intersects the -axis at three points .

       

                            (iv)

Solution : (iv) The number of zeroes is 2 as the graph intersects the -axis at two points .         

                                      

                         (v)

 Solution : (v) The number of zeroes is 4 as the graph intersects the -axis at four points .                

          

                     (vi) 

Solution : (vi) The number of zeroes is 3 as the graph intersects the -axis at three points .            

Exercise 2.2 

1. Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients .

   (i)       

  (ii)       

  (iii)    

  (iv)          

  (v)                

  (vi)   

Solution :  

(i) We have ,             

Let ,

 

So,     

      or   

The zeroes of   are  and    .

The sum of  its zeroes

The product of its zeroes

(ii) We have ,             

Let ,

 

So,     

 

 or   

The zeroes of  are  and   .

The sum of  its zeroes

The product of its zeroes

 (iii)   We have,   

Let ,

 

 

  

 So,     

or   

The zeroes of  are and .

The sum of  its zeroes

The product of its zeroes

  (iv)  We have ,            

Let ,

 

 So,     

  

        or    

The zeroes of   are 0 and 2  .

The sum of  its zeroes

The product of its zeroes

 (v)   Let ,

 

  

 So ,      

         

 or    

 The zeroes of  are  and   .

 The sum of  its zeroes

 The product of its zeroes                 

(vi)  We have ,   

Let ,  

 

So,     

 

       

or   

The zeroes of   are   and    .

The sum of  its zeroes

The product of its zeroes

2. Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively .

(i)         (ii)       (iii)       (iv)      (v)       (vi)   

Solution : (i)  

Let  and  be zeroes of the quadratic polynomial  respectively .

We have ,

                        

and 

If  , then  and

Therefore, the quadratic polynomial is

i.e.,  .

 OR

We know that , 

The quadratic polynomial

 , where  is any real constant .

Therefore, the quadratic polynomial is  .

(ii)   

Let  and  be zeroes of the quadratic polynomial  respectively .

 We have,

                            

 and 

 If  , then   and

Therefore, the quadratic polynomial is  

i.e.,  .

(iii)     

  Let  and  be zeroes of the quadratic polynomial respectively .

We have, 

and 

If  , then  and

Therefore, the quadratic polynomial is  

 i.e.,   .       

(iv)        

 Let  and  be zeroes of the quadratic polynomial  respectively .

We have,

and 

If  , then and

Therefore, the quadratic polynomial is  

i.e.,   .

(v)       

 Let  and  be zeroes of the quadratic polynomial respectively .

 We have,

and 

If  , then and

Therefore, the quadratic polynomial is  

i.e., 

(vi)   

 Let  and  be zeroes of the quadratic polynomial respectively .

 We have,

and 

If  , then and

Therefore, the quadratic polynomial is  

i.e.,  .