Chaper 2 . Polynomials

Class 10 Maths Chapter 2 Polynomials : Multiple Choice Questions , Answer following the Questions , Fill in the blanks , 2 Marks Questions , 3 Marks Question , 4 Marks and Solutions :

 Class 10 Polynomials Multiple Choice Questions  SECTION = A

Q1. If one zeroes of the polynomial  is 1, then value of  is :   [SEBA 2017]

(a)  1                        

(b)  – 1                              

(c)  2                                   

(d)  – 2

Solution:    (a)  1

[ We have,      

 

  

  

      ]

Q2. If the cubic polynomial  , then the sum of its zeroes taken two at a time is :

(a)                            

(b)                              

(c)                                 

(d)  

Solution:   (d)       

[    The sum of its zeroes taken two at a time    .  ]

Q3. If one zero of the quadratic polynomial  is 2 , then the value of  is

(a) 10                            

(b)  – 10                            

(c) 5                                     

(d)  – 5

Solution:  (b)  – 10        

[    let    

     ]

Q4. The product of zeroes of  is: [SEBA 2018]

(a)  4                             

(b) 8                           

(c) 32                        

(d)  0

Solution :  (d)  0

[  The product of zeroes    ]

Q5. A quadratic polynomial , whose zeroes are  – 3  and 4 is

(a)               

(b)              

(c)              

(d) 

Solution:  (c)    

[   Given ,   and 

 

   

         ]

Q6. Which of the following expressions are  polynomial ?

(a)                  

(b)                       

(c)                 

(d)  

Solution : (c)        

[ We have,   is linear polynomial . ]        

Q7.  The product of the zeros of    is      [SEBA 2019]

(a)   – 15                        

(b)  15                               

(c)                            

(d)   

Solution :  (a)   – 15                        

[ The product of zeroes ]

Q8. If the zeroes of the quadratic polynomial   are 2 and – 3 , then

 (a)    

(b)      

(c)    

(d)  

Solution:   (d)  

[  let    

A/Q , Sum of zeroes

And , Product of zeroes

     ]

Q7.  If  ,  and  , then the cubic polynomial is :

(a)                                            

(b)             

(c)                                                            

(d)     

Solution:  (b)      .

[ The cubic polynomial  

 

       ]

Q8.  If the cubic polynomial   , then the sum of the product of its zeroes taken two at a time is :

(a)                       

(b)                        

(c)                        

(d)   

Solution : (d)   

Q9. If   and  is a cubic polynomial , then the number of zeroes of   is :

(a)    0                            

(b)  2                         

(c)  3                         

(d)  4

Solution : (c) 4

Q9. If one of the zeroes of the quadratic polynomial   is  – 3 , then the value of  is :

(a)                                

(b)                           

(c)                                 

(d)  

Solution:   (a)                             

[    let   

 

     ]

Q10. The sum of the zeroes of the cubic polynomial  is :   [SEBA 2015]

(a)  5                  

(b)  11                        

(c)  3                   

(d)   

Solution:  (d)           

[ The sum of the zeroes   ]

Q11. If the graph of the polynomial  intersects  - axis at two points , then number of zeroes of  is : [SEBA 2016]

(a)  0                              

(b)  3                          

(c)  1                        

(d)  2

Solution :  (d) 2 [ The number of zeroes is 2 as the graph intersects the  axis at two points . ]

Q12. If one of the zeroes of the cubic polynomial   is  – 1  , then the product of the other two zeroes is :

(a)                        

(b)                      

(c)                      

(d)  

Solution:  (d)                

[     

 

The product of zeroes

 

         ]

Q13. The zeroes of the quadratic polynomial  are :

(a)  both positive                                                    

(b) both negative

(c)  One positive and one negative                     

(d) both equal

Solution:  (c)  One positive and one negative .                    

[ 

   ]

Q14. If  is a polynomial of at least degree one and , then is know as :

(a) The value of   .       

(b) Zero of   .                  

(c) Constant term of        

(d) none of these

Solution:  (b) Zero of   .                  

Q15. Consider the following statements :

(i)    is a factor of            

(ii)   is a factor of  

(iii)  is a factor of  

In these statements :

(a)  (i) and (ii) are correct                                               

(b)  (i) , (ii) and (iii) are correct 

(c)  (ii) and (iii) are not correct                                        

(d)  (i) and (iii) are correct

Solution:  (c)  (ii) and (iii) are not correct                                        

[ (i)    is a factor of  

 

(ii)   is a factor of  

(iii)   is a factor of   

   ]

Q16. If   , then the value of  is :    [ SEBA 2014]

(a) – 19                 

(b)  – 29                   

(c)  – 39                       

(d)   – 49

Solution:  – 39   

[ We have, 

  ]

Q5. In figure, the graph of a polynomial   is shown , what type polynomial represent in the graph ?

 

(i)  linear polynomial                                                  

(ii) quadratic polynomial

(iii) cubic polynomial                                                  

(iv)  constant polynomial

Solutiion : (d)  (iv)  constant polynomial  

[ because , the graph is not intersecting the  axis .]

Q If   is divisible by   , then the value of   is :  [CBSE 2010]

(i)  7                 

(ii)  8                        

(iii)  9                        

(iv)  10

Solution:  (d) 9    

[ We have ,   and  

 

    ]

Q17. On dividing a polynomial  by , quotient and remainder are found to be and respectively . The polynomial   is  : [ CBSE 2020  standard]

(a)                   

(b)                 

(c)     

(d) 

Solution :  (b)                 

[   divisor  quotient  remainder

   ]

Fill in the blank :

Q1. If 2 is a zero of the polynomial  , then the value of  is  . [CBSE 2020 basic]

Solution:  1

[  let    

  ]

Q2. If  is divisible by , then the value of  and are    and  .

Solution:   2  and   0   .

[ We have,  

let,     

 

 

and      

  ]

Q3. If the sum and product of the zeroes  are  – 3 and 2 , then the polynomial is   .

Solution:    .

[  The polynomial   ]

Q4. If   and  are the zeroes of the quadratic polynomial   , then the sum of zeroes is   . 

Solution:             

[ The sum of zeroes     ]

Q5. If the polynomial  , then the value of  is   .

Solution :   – 1  .   

[ We have,   

   ]

Q6. If   is one of the factors of the polynomials  , then the remaining factor is  .

Solution :   .   

[ We have,  

   ]

Q7.  The coefficient of  in the polynomial  is   .

Solution:   4      

[ We have,    ]

Class 10 Polynomials Answer following the questions :

Q1. If  3 is a zero of the quadratic polynomial , what is the value of  ?  [SEBA 2013]

Solution:  let  

 

  

 

 

Q2.  The graph of is given below , for some polynomial  . Find the number of zeroes of  .           

  

Solution : The number of zeroes is 3 , because the graph intersects the -axis at three points .   

Q3. The graph of   is given in figure , how many zeroes are there of   ?  

 

Solution: The number of zero is 1 , because the graph intersects the -axis at one point only .                      

Q4. If one zero of the polynomial  is  , write other zero .

Solution:  let  other zero is  .

A/Q , The sum of zeroes  

 

  

Q5. Find the quadratic polynomial whose zeros are 3 and – 4 respectively .

Solution:  Here ,   and

  The quadratic polynomial  

 

Q6. If the divisor  , quotient  and remainder  are three polynomial respectively , then find the dividend  of the polynomial .

Solution:  We know that , dividend  divisor  quotient  remainder  .

  

  

  

                            

Class 10 Polynomials 2 Marks Questions and Answers

SECTION = B

Q1. For what value of  ,  is the zero of the polynomial   ? Also, the other zeroes of the polynomial.

Solution:  let    

 

 

 

 

 

So, 

 

 

 

          or       

Thus, the zeroes of the given polynomial are  – 7  and    . 

Q2. Find the zeroes of the quadratic polynomial   .

Solution:  We have, 

   

  

  

      or     

Thus , the zeroes of the quadratic polynomial   are  and   .    

Q3. Find a quadratic polynomial the sum and product of whose zeroes are  – 7 and 10 respectively. Hence find the zeroes .

Solution:  The quadratic polynomial  

 

 

We have ,    

  

  or 

Therefore, the zeroes of    are  – 2 and – 5  .                    

Q4. Find a cubic polynomial with the sum , sum of the product of its zeroes taken two at a time, and the product of its zeroes as 2 , – 7 and  – 14 respectively. 

Solution:  We know that , the cubic polynomial  

 

Therefore , the polynomial is   

Q5. Find a cubic polynomial whose zeroes are  2 , – 4 and  – 3 respectively .

Solution:  Here,   ,   and    .

The cubic polynomial  

 

 

 

Q6. If two zeroes of the polynomial  are  and  ,then find its third zero. [CBSE 2010]                                                                                                                                 

Solution:  let  be the third zero .

Here ,  and   .

The sum of the zeroes  

 

 

Q7. Find the quadratic polynomial whose zeroes are  and    .

Solution: Here,   

 and    

The quadratic polynomial  

 

 

 

 

   

Q8. If  and  are the zeroes of the polynomial  , find the value of  . [Delhi 2013]

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

        and             

Now , 

  

Class 10 Polynomials 3 Marks Questions and Answers

SECTION = C

Q1. Divide the polynomial  by the polynomial  and find the quotient and the remainder :

     and       [SEBA 2019]

Solution:  Given,     and      

Now

 

 Thus , the quotient and the remainder  

Q2. Find the zeroes of the polynomial  and verify the relationship between the zeroes and the coefficients .

Solution:  We have, 

 

 

      or       

Therefore , the zeroes of the given polynomial  and    .

The sum of the zeroes  

The product of the zeroes   verified .

Q3. Verify that  4 , – 2  ,    are the zeroes of the cubic polynomial   and then verify the relationship between the zeroes and the coefficients .

Solution:  let  

 

 

 

Here ,   ,     and 

So,     

and  

    

Q4. Find the quadratic  polynomial whose zeroes are  – 2 and  – 5 . Verify the relationship between zeroes and coefficients of the polynomial. [ Delhi 2013  , SEBA 2018]  

Solution:   Here,   and  

The quadratic polynomial  , where  is a constant .

 

Therefore, the quadratic polynomial is  .

 Now ,   the sum of zeroes  

 The product of zeroes    verified.

Q5. If the zeroes of the polynomial  are ,  , , find  and  .

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

A/Q , the sum of the zeroes

 

And  product of zeroes  

 

 

 

 

 

 

 

Therefore , the value of and  are 1 and  .

Q7. If  and  are the zeroes of the polynomial , then form a quadratic polynomial whose zeroes are  and

Solution:  We have,   

    and     

The sum of zeroes   

The product of zeroes   

The quadratic polynomial  ,  where  is constant .

 

 

 

Therefore , the quadratic polynomial is  .  

Q8. Verify that  3 ,  – 1 ,  –   are the zeroes of the cubic polynomial    and then verify the relationship between the zeroes and the coefficients .

Solution:  let    and given, 3 ,  – 1 ,  –   are the zeroes of  .

   

        0  

We take ,  ,  and    

    

Therefore ,  3 , – 1 and  –   are the zeroes of the cubic polynomial of   .

Class 10 Polynomials 4 Marks Questions and Answers

SECTION = D

Q1. If the zeroes of the polynomial   are  , and  , find and as well as the zeroes of the given polynomial.

Solution:  The polynomial is   

The sum of  its zeroes

The product of its zeroes  

   

 

  [ From  ]

 

 

Putting  the value of   and  in equation  , we get

          and   

Therefore, the value of  and  are  or  and  or  .