2 : Polynomials

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Importants Notes :

1. Polynomials : A polynomial in one variable  of degree  is an expression of the form
 , where  are constants and  .

2. A polynomials having only one term are called monomials .

Example :  etc .

3. A polynomials having only two terms are called binomials .

Example :  etc .

4. A polynomials having only three terms are called trinomials .

Example : etc .

5. The degree of a non-zero constant polynomial is zero .

6. A polynomial of degree one is called a linear polynomial. The general form of a linear polynomial is  Where  and  are constants and. Example :  , …….. etc

7. A polynomial of degree two is called a quadratic polynomial. The general form of a quadratic polynomial is  Where  and  are constants and. Example :   …….. etc

8. A polynomial of degree three is called a cubic polynomial. The general form of a cubic polynomial is  Where  and  are constants and.

Example :  , …….. etc .

9. The constant polynomial 0 is called the zero polynomial . Example : 2 , – 5 ,  , …… etc .

9 . Factor Theorem :  is a factor of the polynomial  if . Also, if  is a factor of  then p .

10. Remainder Theorem : If  is any polynomial of degree greater than or equal to 1 and  is divided by the linear polynomial , then the remainder is

11. Dividend = (Divisor × Quotient) + Remainder .

12. Algebraic Identities :

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)  

(viii)  

(ix) If  , then z .

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EXERCISE = 2.1

1. Which of the following expressions are polynomials in one variables and which are not ? State reasons for your answer .

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

Solution :  (i)   is a polynomial in the variable  .      

(ii)  is a polynomial in the variable  .  

(iii)   is not a polynomial in the variable  . 

(iv)  is not a polynomial in the variable  .  

(v)   is not a polynomial in the three variable .

2. Write the coefficients of  in each of the following :

 (i)       (ii)      (iii)          (iv)  

Solution: (i)    

 The coefficients of   is 1 .

 (ii)     

 The coefficients of  is  .

 (iii)      

The coefficients of   is   .

   (iv)   

 The coefficients of  is 0 .

3. Give one example each of a binomial of degree 35 and of a monomial of degree 100 .

Solution : The example of a binomial of degree 35 is  and  the example of a monomial of degree 100 is  .

4. Write the degree of each of the following polynomials :

  (i)          (ii)        (iii)              (iv)

Solution : (i)      

 The highest power of the variable is 3 .

So , the degree of the polynomial is 3 .

  (ii)      

 The highest power of the variable is 2 .

So , the degree of the polynomial is 2 .

   (iii)   

 The highest power of the variable is 1 .

 So , the degree of the polynomial is 1 .

  (iv)    

The exponent of  is 0 .

So , the degree of the polynomial is 0 .

5. Classify the following as linear , quadratic and cubic polynomials :

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

Solution :  (i)   is a quadratic polynomial .  

  (ii)   is a cubic polynomial .

 (iii)  is a quadratic polynomial .

   (iv) is a linear polynomial .

  (v)  is a linear polynomial .

 (vi)  is a linear polynomial .

 (vii)   is a quadratic polynomial .

EXERCISE 2.2

1. Find the value of the polynomial  at   (i)          (ii)          (iii) 

Solution : Let  

  (i) Given,                                                               

   

(ii) Given ,       

 

       

(iii) Given ,

  

   

2. Find  and  for each of the following polynomials :

  (i)     

 (ii)     

 (iii)   

(iv)

Solution : (i)    

   

   

     

 (ii)  

   

   

     

 (iii)  

 

 

 

(iv) 

 

 

   

3. Verify whether the following are zeroes of the polynomial , indicated against them .

(i) 

(ii)

(iii)

(iv)

(v)

(vi)

(vii) 

(viii)  

Solution:  (i) We have,

 ,

Therefore, is the zero of the polynomial  .

Solution : (ii) We have,

, 

Therefore,  is not the zero of the polynomial  .

Solution:  (iii) We have , ,

 

And  

Therefore,  and  are the zeroes of the polynomial  .

Solution:  (iv) We have,  ,

 

 

And   

 

Therefore,  and   are the zeroes of the polynomial  .

Solution:  (v) We have

 

Therefore,  is the zero of the polynomial  .

Solution: (vi) We have,  

 ,

Therefore, is the zero of the polynomial  .

 Solution: (vii) We have,

   ,

Therefore,  is the zero of the polynomial  .

 

Therefore,    is not the zero of the polynomial  .

 Solution:  (viii) We have,  ,

Therefore, is not the zero of the polynomial  .

4. Find the zero of the polynomial in each of the following case

   (i) 

 (ii)

 (iii) 

 (iv)

 (v)

 (vi)

(vii)  are real numbers .

Solution :  (i) We have, 

 

So,  – 5 is the zero of  .

Solution: (ii) We have, 

 

 

So,  5 is the zero of  .

Solution: (iii)We have, 

 

 

So, is the zero of  .

Solution: (iv) We have,  

 

 

 

So, is the zero of  .

 Solution: (v) We have, 

 

 

So,  0 is the zero of  .

Solution: (vi) We have,

 

 

So,  0 is the zero of  .

Solution: (vii) We have,  are real numbers .

 

 

So, is the zero of  .

EXERCISE 2.3

1. Find the remainder when  is divided by

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

Solution: (i)  Let   and the zero of   is  – 1 .

 

 

     

Therefore, the remainder is 0 .

(ii)  Let   and the zero of is 

Therefore, the remainder is  .

(iii)  Let  and the zero of   is   .

 

 

Therefore, the remainder is  .

(iv) Let  and the zero of   is  0 .

 

 

Therefore, the remainder is 1 .

(v) Let  and the zero of  is

Therefore, the remainder is

2. Find the remainder when  is divided by   .

Solution:  let  and the zero of   is  .

 

 

Therefore, the remainder is  .

3. Check whether is a factor of  .

Solution: Let  and the zero of  is

Therefore, is not a factor of   .

EXERCISE 2.4

1 Determine which of the following polynomials  has( ) a factor :

 (i)                                                     

(ii)     

 (iii)                                     

(iv)  

Solution : (i) Let   

If  is factor of  then, the zero of  is  – 1  .

  

              

                

Therefore,  is a factor of  .                                           

(ii)     

If  is factor of  then, the zero of  is  – 1  .

  

              

                

Therefore,  is not a factor of p .                                           

 (iii) Let       

If  is factor of  then, the zero of  is  – 1  .

            

            

Therefore,  is not a factor of p .                                                                           

(iv) Let   

If  is factor of  then, the zero of  is  – 1  .

 

   

   

Therefore,  is not a factor of p .                                                                          

2. Use the Factor Theorem to determine whether  is a factor of  in each of the following cases:

(i)

(ii)

(iii) 

Solution :

(i)

If  is factor of  then, the zero of  is  – 1  .

 

             

             

             

Therefore,  is a factor of p .

(ii)  

If  is factor of  then, the zero of  is  – 2  .

 

             

             

             

Therefore,  is not a factor of p .

(iii)  

If   is factor of  then, the zero of  is 3 .

    

             

              

             

Therefore,  is a factor of p .

3. Find the vaule of  if  is a factor of  in each of the following cases :

(i)                    

(ii)

(iii)        

(iv)  

Solution : (i) We have,    

Since  is factor of p .

So , the zero of  is 1 .

 

 

 

               

Therefore, the value of k is - 2 .

 (ii) We have,

Since  is factor of p .

 So , the zero of  is 1 .

 

               

(iii) We have,   

Since  is factor of p .

So , the zero of  is 1 .

 

 

 

      

(iv) We have,  

Since  is factor of p .

So , the zero of  is 1 .

 

4. factorise:

(i)                             

(ii)   

(iii)                             

(iv)

Solution : (i) We have,

 

 

                           

(ii) We have,  

 

 

  

(iii) We have,  

 

 

                            

(iv) We have,  

  

 

 

5. Factorise:

(i)                         

(ii)

(iii)             

(iv)     

Solution : (i) We have,  

  

 

                       

(ii)We have,  

 

  

 

 

 

  

  

(iii) We have,       

 

 

  

 

 

 

(iv) We have,  

 

 

     

EXERCISE - 2.5

1.  Use suitable identities to find the following  products:

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

Solution : (i) We have, 

 

                      

 (ii) We have, 

 

         

 (iii) We have, 

 

 

(iv) We have, 

 

                  

(v) We have, 

 

        

2. Evaluate the following products without multiplying directly:

(i)    (ii) 95×96       (iii) 104×96    

Solution : (i) We have,

 

 

 

 

 

 (ii) We have,  95×96

 

 

 

    

 (iii) We have,

 

 

 

  

3. Factorise the following using appropriate identities:

 (i)     (ii)       (iii) 

Solution : (i) We have, 

 

 

              

(ii) We have,  

 

 

          

 (iii) We have, 

 

 

4 . Expand  each  of the following ,using suitable identities :

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

Solution : (i) We have,

 

                

(ii) We have,        

 

                

(iii) We have,   

 

                

(iv) We have,  

 

                                

(v) We have,   

                       

(vi) We have, 

 

  

   

5. Factorise:

(i)  

(ii) 

Solution : (i) We have, 

 

 

 

(ii) We have, 

 

 

 

6. Write the following cubes in expanded form:

(i)        (ii)      (iii)      (iv)

Solution : (i) We have,

 

       

(ii) We have,

 

  

(iii) We have,  

 

    

(iv) We have,  

 

 

7. Evaluate the following using suitable identitites:

(i)     (ii)       (iii)

Solution : (i) We have,

 

 

 

 

 

              

(ii) We have

 

 

 

 

                            

(iii) We have,

 

 

 

 

                            

8.Factories each of the following :

(i)             

(ii) 

(iii)        

(iv)  

(v)  

Solution :

(i) We have,  

 

 

           

(ii) We have,   

 

 

           

(iii) We have,  

 

 

      

(iv) We have,  

  

 

 

(v) We have, 

   

 

9. Verify:

(i)             

(ii)  

Solution:  (i)   

R.H.S. : 

 

 

  L.H.S.  verified.            

 (ii)  

R.H.S. : 

 

 

  L.H.S.  verified.            

10. Factorise each of the following:

(i)         (ii) 

Solution : (i) We have,

  

 

        

(ii) We have, 

 

 

  

11.Factorise:   

Solution : We have,

 

 

 

12.Verify that:

Solution :  We have ,

 

   

    

  

  verified .

13. If show that  . .

Solution : we have ,

 

 

 

  

       

14. Without actually calculating the cubes , find the value of each of the following :

(i)                    

(ii)   

Solution : (i) We have,                    

Solution :  Here ,  , b=7,  

  

   [ ]

  

 

     

(ii) We have,   

 Here ,   ,  ,  

    

 [ ]

  

 

      

15. Give the possible expression for the length and breadth of each of the following rectangles ,in which their areas are given :

Area:

                         (i)

Area :

                    (ii)

Solution : (i) We have,

 

 

 

(ii) We have,

 

 

 

16. What  are the possible expreesions for the dimensions of the cuboids whose volume are given bellow ?

Volume :

                   (i)

Volume :

                          (ii)                            

Solution : (i) We have,

     

           

(ii) We have,