Chapter 2. POLYNOMIALS 

Chapter 2 : Polynomials

Exercise 2.1 Complete Solution Exercise 2.2 Complete Solution Exercise 2.3 Complete Solution Exercise 2.4 (Optional) 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

Division Algorithm for Polynomials : 4. Dividend = Divisor × Quotient + Remainder 5. If  and  are any two polynomials with , then we can find polynomials  and  such that  where  or degree of  degree of This result is known as the Division Algorithm for polynomials.

Class 10 Maths Chapter 2. POLYNOMIALS Exercise 2.1 Solutions :

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 .            

 Class 10 Maths Chapter 2. POLYNOMIALS Exercise 2.2 Solutions  

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.,  .

Class 10 Maths Chapter 2. POLYNOMIALS Exercise 2.3 Solutions : 

1. Divide the polynomials  by the polynomial  and find the quotient and remainder in each of the following :

 (i)    ,    

(ii)     ,   

(iii)     ,    

Solution :  (i) We have,

    ,   

 Now ,          

     

Therefore, the quotient is  and the remainder is  .

(ii) We have ,

     , 

  

  Now ,     

    

  Therefore, the quotient is   and the remainder is 8 .

 (iii) We have,

     ,  

   Now ,

  

Therefore, the quotient is   and the remainder is   .

2. Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial :

 (i)   

(ii) 

(iii) 

Solution : (i) We have,   

  Now ,

   

 Therefore, the polynomial  is a factor of the polynomial   .

(ii) We have,

 

  Now ,  
    

Therefore, the polynomial  is a factor of   .

(iii) We have, 

 Now ,

 

Therefore , the polynomial  is a factor of polynomial .

3. Obtain all other zeroes of  , if two of its zeroes are and .

Solution : Let  

 Since two zeroes are  and .

So,

is a factor of  .

  Now , 

  

  

 

    

  ;    ;   or 

Therefore , the zeroes of the given polynomial are  and  .

4. On dividing  by a polynomial , the quotient and remainder were  and , respectively . Find   .

Solution :  Let ,  ,   ,       

     

  

  

     

  Now ,

   

      

5. Give example of polynomials p(x) , g(x) , q(x) and r(x) , which satisfy the division algorithm and

 (i)       

 (ii)         

 (iii)   .

Solution : (i)   Let     ; 

 

(ii)  Let   ;    ;     ;    

 

(iii)  Let   ;   ;    ; 

   

6. (i) If one zero of the polynomial  is 1 , then find all the other zeros .

Solution : Let

Since, 1 is the zero of the polynomial   .

Now ,

   

 

 ,  or

,  or

Therefore , the zeroes of the given polynomial are 1 , – 1 and .

(ii) If one zero of the polynomial  are and  , then find all the other zeros .

Solution : let

Since and   are the zeroes of the polynomials and  .

 is the factor of  .

Now

     

 

   ,   or

  , or 

Therefore , the zeroes of the given polynomial are ,   , – 3 and  2 .

(iii) If one zero of the polynomial  are and  , then find all the other zeros .

Solution :  let 

Since and  are the zeroes of the polynomials and  .

 is the factor of  .

Now

  

 

 ,  ,   or   

  ,  ,  or

Therefore , the zeroes of the given polynomial are ,   , 1 and 1 .

7. (i) On dividing the polynomial  by another polynomial the remainder is found as – 15 . Find the quotient .

Solution : Given, dividend

Let

Now

  

Therefore, the quotient is   .

(ii) On dividing a polynomial by , the quotient is found as and the remainder as  . Find the polynomial .

Solution : Given,  , and

We know that ,  Dividend = divisor  quotient + remainder 

 

Therefore, the polynomial is  .

Class 10 Maths Chapter 2. POLYNOMIALS Exercise 2.4 Solutions (Optional)

1. Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also verify the relationship between the zeroes and the coefficients in each case :

  (i)    (ii) 

Solution : (i)  let ,   : and 

Again ,

 

 

 

 

Therefore, , 1 and  are the zeroes of   .

Let ,  and

The sum of zeroes

The sum the product of its zeroes taken two at a time

The product of zeroes

   Verified .

(ii) let ,    and  and 1

  

Again ,  

  

And  

    

Therefore , 2 , 1 and 1 are the zeroes of  .

Let ,  and

The sum of zeroes

The sum the product of its zeroes taken two at a time

The product of zeroes

   Verified .

2. Find a cubic polynomial with the sum , sum of the product of its zeroes take two at a time and the product of its zeroes as  respectively .

Solution :

Solution: let ,  ,  and  are the zeroes of the cubic polynomial  respectively .

 

   

 

and

If  , then  and  

So , the quadratic polynomial is

3. If the zeroes of the polynomial   and  ; find  and  .

Solution :  Since, and are the zeroes of the polynomial  .

The sum of zeroes

The product of zeroes =

Therefore , the value of  and .

4. If two zeroes of the polynomial  are  , find other zeroes .

Solution: We have ,  

Since two zeroes are   and  .

So ,   

 is the factor of  .

Now ,  
      

Quotient   and Remainder

 

 

 

 

The factor of   are  and   .

So, its zeroes are 7 and  – 5 .

Therefore, the zeroes of the given polynomial are   ,   , 7 and – 5 .  

5. If the polynomial  is divided by another polynomial  , the remainder comes out to be , find  and  .

Solution :  We  have ,

Dividend  

Divisor

Now ,

Remainder  

A/Q, 

Comparing the coefficients  and constant terms on both the sides , we get

   

 

 

   

and  

   [putting  ]

 

 

 

Therefore, the value of  and  .