07 - Question bank - Polynomials - Class 10 - Maths Class 10 Notes | EduRev

Crash Course for Class 10 Maths by Let's tute

Class 10 : 07 - Question bank - Polynomials - Class 10 - Maths Class 10 Notes | EduRev

 Page 1


   POLYNOMIALS
 
 
 
GRAPHS OF POLYNOMIALS 
 
1) Draw a graph of the polynomial f x x
    Solution: 
 Let y = 3 2 x+ now we get different values of y corresponding to the different values of x. 
  
x 0 1 2 
 y 2 5 8 
 
 The points A(0,2), B(1,5) and C(2,8) are plotted on the graph paper on a suitable scale.
 The points are joint together to get the graph of the given polynomial. 
Since the polynomial is linear we get the graph as a straight line. 
 
          
 
RELATIONSHIP BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL
2) Find the zeroes of the quadratic polynomial 
     between the zeroes and the coefficients.
     Solution:  
     The given quadratic polynomial can be
     Thus the value of 
2
12 32 x x + + is zero when 
 i.e. when 8 x=- or 4 x=- Therefore the zeroes of 
 Sum of zeroes = -8+(-4)=-12 Product of the zeroes=
   
POLYNOMIALS 
( ) 3 2 f x x = + 
now we get different values of y corresponding to the different values of x. 
 
 
points A(0,2), B(1,5) and C(2,8) are plotted on the graph paper on a suitable scale.
The points are joint together to get the graph of the given polynomial.  
Since the polynomial is linear we get the graph as a straight line.  
BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL
Find the zeroes of the quadratic polynomial 
2
12 32 x x + + and verify the relationship 
between the zeroes and the coefficients. 
The given quadratic polynomial can be factorised as 
2
12 32 ( 8)( 4) x x x x + + = + +
is zero when ( 8) 0 x+ = or ( 4) 0 x+ = 
Therefore the zeroes of 
2
12 32 x x + + are -8 and -4. 
12 Product of the zeroes=-8 x -4=32 
now we get different values of y corresponding to the different values of x.  
points A(0,2), B(1,5) and C(2,8) are plotted on the graph paper on a suitable scale. 
 
BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL  
and verify the relationship  
12 32 ( 8)( 4) x x x x + + = + + 
4.  
Page 2


   POLYNOMIALS
 
 
 
GRAPHS OF POLYNOMIALS 
 
1) Draw a graph of the polynomial f x x
    Solution: 
 Let y = 3 2 x+ now we get different values of y corresponding to the different values of x. 
  
x 0 1 2 
 y 2 5 8 
 
 The points A(0,2), B(1,5) and C(2,8) are plotted on the graph paper on a suitable scale.
 The points are joint together to get the graph of the given polynomial. 
Since the polynomial is linear we get the graph as a straight line. 
 
          
 
RELATIONSHIP BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL
2) Find the zeroes of the quadratic polynomial 
     between the zeroes and the coefficients.
     Solution:  
     The given quadratic polynomial can be
     Thus the value of 
2
12 32 x x + + is zero when 
 i.e. when 8 x=- or 4 x=- Therefore the zeroes of 
 Sum of zeroes = -8+(-4)=-12 Product of the zeroes=
   
POLYNOMIALS 
( ) 3 2 f x x = + 
now we get different values of y corresponding to the different values of x. 
 
 
points A(0,2), B(1,5) and C(2,8) are plotted on the graph paper on a suitable scale.
The points are joint together to get the graph of the given polynomial.  
Since the polynomial is linear we get the graph as a straight line.  
BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL
Find the zeroes of the quadratic polynomial 
2
12 32 x x + + and verify the relationship 
between the zeroes and the coefficients. 
The given quadratic polynomial can be factorised as 
2
12 32 ( 8)( 4) x x x x + + = + +
is zero when ( 8) 0 x+ = or ( 4) 0 x+ = 
Therefore the zeroes of 
2
12 32 x x + + are -8 and -4. 
12 Product of the zeroes=-8 x -4=32 
now we get different values of y corresponding to the different values of x.  
points A(0,2), B(1,5) and C(2,8) are plotted on the graph paper on a suitable scale. 
 
BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL  
and verify the relationship  
12 32 ( 8)( 4) x x x x + + = + + 
4.  
                                       POLYNOMIALS
 
 
 
 
 
         
( )
( )
2
Coefficient of x
Coefficient of x
-
= 
(12)
1
-
= =-
  
(
(
Coefficient of x
Sum of zeroes=   Product of the zeroes=
Coefficient of x
-
?
 
3) Find the zeroes of the quadratic polynomial 
between the zeroes and the coefficients.
     Solution:      
Using the identity 
2 2
( )( ) a b a b a b - = + -
                  ?

v6

	
Hence, the value of 

v6

  is zero when 
Hence x = -v6 and x = v6 are the zeroes of 
Sum of zeroes 6 6 0 = - =  Product of zeroes
( )
( )
2
Coefficient of x
Coefficient of x
-
0
0
1
= = as the quadratic polynomial x
 
2
Constant term
Coefficient of x
6
6
1
-
= =- 
(
(
Coefficient of x
Sum of zeroes=   Product of the zeroes=
Coefficient of x
-
?
 
4) Find a quadratic polynomial, the sum and prod
 Solution; 
 Let the quadratic polynomial be ax bx c
 Given that  
   6 a ß + =-
b
a
-
= b a ? =
 and       x 5
c
a
a
ß = =    5 c a ? =
 If a=1, then b=6 and c=5  
 Thus, quadratic polynomial which satisfies the given condition is 
 
 
POLYNOMIALS 
(12)
12 = =-    
2
Constant term
Coefficient of x
 
32
32
1
= =
 
)
)
2
Coefficient of x
Constant term
Sum of zeroes=   Product of the zeroes=
Coefficient of x
Coefficient of x
Find the zeroes of the quadratic polynomial 
2
6 x - and verify the relationship  
between the zeroes and the coefficients. 
( )( ) a b a b a b - = + - we can write, 
v6		v6 
is zero when 6 0 x+ = or 6 0 x- = i.e. x=-
6 are the zeroes of the quadratic polynomial x
2
 - 6 
Product of zeroes 6 x - 6 6 = =- 
as the quadratic polynomial x
2
 – 6 we can write it as x
)
)
2
Coefficient of x
Constant term
Sum of zeroes=   Product of the zeroes=
Coefficient of x
Coefficient of x
Find a quadratic polynomial, the sum and product of whose zeroes are -6 and 5
2
ax bx c + + and its zeroes be a and ß 
6 b a ? = 
5 c a ? =  
quadratic polynomial which satisfies the given condition is 
2
6 5 x x + + 
2
Constant term
Coefficient of x
  
 
6 x=- or 6 x=
 
write it as x
2
 + 0x - 6 
2
Constant term
Coefficient of x
 
6 and 5 respectively   
6 5 
Page 3


   POLYNOMIALS
 
 
 
GRAPHS OF POLYNOMIALS 
 
1) Draw a graph of the polynomial f x x
    Solution: 
 Let y = 3 2 x+ now we get different values of y corresponding to the different values of x. 
  
x 0 1 2 
 y 2 5 8 
 
 The points A(0,2), B(1,5) and C(2,8) are plotted on the graph paper on a suitable scale.
 The points are joint together to get the graph of the given polynomial. 
Since the polynomial is linear we get the graph as a straight line. 
 
          
 
RELATIONSHIP BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL
2) Find the zeroes of the quadratic polynomial 
     between the zeroes and the coefficients.
     Solution:  
     The given quadratic polynomial can be
     Thus the value of 
2
12 32 x x + + is zero when 
 i.e. when 8 x=- or 4 x=- Therefore the zeroes of 
 Sum of zeroes = -8+(-4)=-12 Product of the zeroes=
   
POLYNOMIALS 
( ) 3 2 f x x = + 
now we get different values of y corresponding to the different values of x. 
 
 
points A(0,2), B(1,5) and C(2,8) are plotted on the graph paper on a suitable scale.
The points are joint together to get the graph of the given polynomial.  
Since the polynomial is linear we get the graph as a straight line.  
BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL
Find the zeroes of the quadratic polynomial 
2
12 32 x x + + and verify the relationship 
between the zeroes and the coefficients. 
The given quadratic polynomial can be factorised as 
2
12 32 ( 8)( 4) x x x x + + = + +
is zero when ( 8) 0 x+ = or ( 4) 0 x+ = 
Therefore the zeroes of 
2
12 32 x x + + are -8 and -4. 
12 Product of the zeroes=-8 x -4=32 
now we get different values of y corresponding to the different values of x.  
points A(0,2), B(1,5) and C(2,8) are plotted on the graph paper on a suitable scale. 
 
BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL  
and verify the relationship  
12 32 ( 8)( 4) x x x x + + = + + 
4.  
                                       POLYNOMIALS
 
 
 
 
 
         
( )
( )
2
Coefficient of x
Coefficient of x
-
= 
(12)
1
-
= =-
  
(
(
Coefficient of x
Sum of zeroes=   Product of the zeroes=
Coefficient of x
-
?
 
3) Find the zeroes of the quadratic polynomial 
between the zeroes and the coefficients.
     Solution:      
Using the identity 
2 2
( )( ) a b a b a b - = + -
                  ?

v6

	
Hence, the value of 

v6

  is zero when 
Hence x = -v6 and x = v6 are the zeroes of 
Sum of zeroes 6 6 0 = - =  Product of zeroes
( )
( )
2
Coefficient of x
Coefficient of x
-
0
0
1
= = as the quadratic polynomial x
 
2
Constant term
Coefficient of x
6
6
1
-
= =- 
(
(
Coefficient of x
Sum of zeroes=   Product of the zeroes=
Coefficient of x
-
?
 
4) Find a quadratic polynomial, the sum and prod
 Solution; 
 Let the quadratic polynomial be ax bx c
 Given that  
   6 a ß + =-
b
a
-
= b a ? =
 and       x 5
c
a
a
ß = =    5 c a ? =
 If a=1, then b=6 and c=5  
 Thus, quadratic polynomial which satisfies the given condition is 
 
 
POLYNOMIALS 
(12)
12 = =-    
2
Constant term
Coefficient of x
 
32
32
1
= =
 
)
)
2
Coefficient of x
Constant term
Sum of zeroes=   Product of the zeroes=
Coefficient of x
Coefficient of x
Find the zeroes of the quadratic polynomial 
2
6 x - and verify the relationship  
between the zeroes and the coefficients. 
( )( ) a b a b a b - = + - we can write, 
v6		v6 
is zero when 6 0 x+ = or 6 0 x- = i.e. x=-
6 are the zeroes of the quadratic polynomial x
2
 - 6 
Product of zeroes 6 x - 6 6 = =- 
as the quadratic polynomial x
2
 – 6 we can write it as x
)
)
2
Coefficient of x
Constant term
Sum of zeroes=   Product of the zeroes=
Coefficient of x
Coefficient of x
Find a quadratic polynomial, the sum and product of whose zeroes are -6 and 5
2
ax bx c + + and its zeroes be a and ß 
6 b a ? = 
5 c a ? =  
quadratic polynomial which satisfies the given condition is 
2
6 5 x x + + 
2
Constant term
Coefficient of x
  
 
6 x=- or 6 x=
 
write it as x
2
 + 0x - 6 
2
Constant term
Coefficient of x
 
6 and 5 respectively   
6 5 
  POLYNOMIALS
 
 
 
 
5) Verify that 2,-3 and 6 are the zeroes of the cubic polynomial 
 and then verify the relationship between the zeroes and the coefficients.
 Solution:  
 Compare the given polynomial with 
 a=1 b=-5 c=-12 and d=36. 
     Now to verify if 2,-3 and 6  are the roots of the polynomial.
     i.e. we have to show that 2	
     ? 
3 2
(2) 2 5(2) 12(2) 36 8 20 24 36 0 f = - - + = - - + =
 
3 2
( 3) ( 3) 5( 3) 12( 3) 36 27 45 36 36 0 f - = - - - - - + =- - + + =
 
3 2
(6) (6) 5(6) 12(6) 36 216 180 72 36 0 f = - - + = - - + =
 Therefore 2,-3 and 6 are the zeroes of the cubic polynomial 
 Let    =2 a  3 ß =- and 6 ? = 
  + 2 ( 3) 6 2 3 6 5 a ß ? + = + - + = - + = =
    2( 3) ( 3)6 6(2) 6 18 12 12 a a ß ß? ? + + = - + - + =- - + =- =
     2( 3)6 36
d
a
a
ß?
-
= - =- =  
 
6) If a and ß are the zeroes of the polynomial 
 Solution: 
 We know that  
(
(
Sum of zeroes=
Coefficient of x
-
               Product of the zeroes=
Coefficient of x
                   And hence,  
1 1
a ß aß
+ =
                                                     ?
 
 
 
 
POLYNOMIALS 
3 and 6 are the zeroes of the cubic polynomial 
3 2
( ) 5 12 36 f x x x x = - - +
elationship between the zeroes and the coefficients. 
Compare the given polynomial with 
3 2
ax bx cx d + + + we get, 
are the roots of the polynomial. 
 0 , 3 = 0 and 6  = 0 
(2) 2 5(2) 12(2) 36 8 20 24 36 0 = - - + = - - + = 
( 3) ( 3) 5( 3) 12( 3) 36 27 45 36 36 0 - = - - - - - + =- - + + = 
(6) (6) 5(6) 12(6) 36 216 180 72 36 0 = - - + = - - + = 
3 and 6 are the zeroes of the cubic polynomial 
3 2
5 12 36 x x x - - +
 + 2 ( 3) 6 2 3 6 5
b
a
-
+ = + - + = - + = = 
2( 3) ( 3)6 6(2) 6 18 12 12
c
a
+ + = - + - + =- - + =- = 
  
are the zeroes of the polynomial 
2
( ) f x ax bx c = + + then find 
1 1
a
+
( )
)
2
Coefficient of x
Coefficient of x
-
 
    i.e. 
b
a
a
ß
-
+ = 
2
Constant term
Coefficient of x
           i.e.   
c
a
a
ß = 
a ß
a ß aß
+
+ = 
b
a
c
a
-
=
b
c
-
= 
? 
1 1 b
c a ß
-
+ = 
( ) 5 12 36 f x x x x = - - + 
5 12 36  
1 1
ß
+ 
Page 4


   POLYNOMIALS
 
 
 
GRAPHS OF POLYNOMIALS 
 
1) Draw a graph of the polynomial f x x
    Solution: 
 Let y = 3 2 x+ now we get different values of y corresponding to the different values of x. 
  
x 0 1 2 
 y 2 5 8 
 
 The points A(0,2), B(1,5) and C(2,8) are plotted on the graph paper on a suitable scale.
 The points are joint together to get the graph of the given polynomial. 
Since the polynomial is linear we get the graph as a straight line. 
 
          
 
RELATIONSHIP BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL
2) Find the zeroes of the quadratic polynomial 
     between the zeroes and the coefficients.
     Solution:  
     The given quadratic polynomial can be
     Thus the value of 
2
12 32 x x + + is zero when 
 i.e. when 8 x=- or 4 x=- Therefore the zeroes of 
 Sum of zeroes = -8+(-4)=-12 Product of the zeroes=
   
POLYNOMIALS 
( ) 3 2 f x x = + 
now we get different values of y corresponding to the different values of x. 
 
 
points A(0,2), B(1,5) and C(2,8) are plotted on the graph paper on a suitable scale.
The points are joint together to get the graph of the given polynomial.  
Since the polynomial is linear we get the graph as a straight line.  
BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL
Find the zeroes of the quadratic polynomial 
2
12 32 x x + + and verify the relationship 
between the zeroes and the coefficients. 
The given quadratic polynomial can be factorised as 
2
12 32 ( 8)( 4) x x x x + + = + +
is zero when ( 8) 0 x+ = or ( 4) 0 x+ = 
Therefore the zeroes of 
2
12 32 x x + + are -8 and -4. 
12 Product of the zeroes=-8 x -4=32 
now we get different values of y corresponding to the different values of x.  
points A(0,2), B(1,5) and C(2,8) are plotted on the graph paper on a suitable scale. 
 
BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL  
and verify the relationship  
12 32 ( 8)( 4) x x x x + + = + + 
4.  
                                       POLYNOMIALS
 
 
 
 
 
         
( )
( )
2
Coefficient of x
Coefficient of x
-
= 
(12)
1
-
= =-
  
(
(
Coefficient of x
Sum of zeroes=   Product of the zeroes=
Coefficient of x
-
?
 
3) Find the zeroes of the quadratic polynomial 
between the zeroes and the coefficients.
     Solution:      
Using the identity 
2 2
( )( ) a b a b a b - = + -
                  ?

v6

	
Hence, the value of 

v6

  is zero when 
Hence x = -v6 and x = v6 are the zeroes of 
Sum of zeroes 6 6 0 = - =  Product of zeroes
( )
( )
2
Coefficient of x
Coefficient of x
-
0
0
1
= = as the quadratic polynomial x
 
2
Constant term
Coefficient of x
6
6
1
-
= =- 
(
(
Coefficient of x
Sum of zeroes=   Product of the zeroes=
Coefficient of x
-
?
 
4) Find a quadratic polynomial, the sum and prod
 Solution; 
 Let the quadratic polynomial be ax bx c
 Given that  
   6 a ß + =-
b
a
-
= b a ? =
 and       x 5
c
a
a
ß = =    5 c a ? =
 If a=1, then b=6 and c=5  
 Thus, quadratic polynomial which satisfies the given condition is 
 
 
POLYNOMIALS 
(12)
12 = =-    
2
Constant term
Coefficient of x
 
32
32
1
= =
 
)
)
2
Coefficient of x
Constant term
Sum of zeroes=   Product of the zeroes=
Coefficient of x
Coefficient of x
Find the zeroes of the quadratic polynomial 
2
6 x - and verify the relationship  
between the zeroes and the coefficients. 
( )( ) a b a b a b - = + - we can write, 
v6		v6 
is zero when 6 0 x+ = or 6 0 x- = i.e. x=-
6 are the zeroes of the quadratic polynomial x
2
 - 6 
Product of zeroes 6 x - 6 6 = =- 
as the quadratic polynomial x
2
 – 6 we can write it as x
)
)
2
Coefficient of x
Constant term
Sum of zeroes=   Product of the zeroes=
Coefficient of x
Coefficient of x
Find a quadratic polynomial, the sum and product of whose zeroes are -6 and 5
2
ax bx c + + and its zeroes be a and ß 
6 b a ? = 
5 c a ? =  
quadratic polynomial which satisfies the given condition is 
2
6 5 x x + + 
2
Constant term
Coefficient of x
  
 
6 x=- or 6 x=
 
write it as x
2
 + 0x - 6 
2
Constant term
Coefficient of x
 
6 and 5 respectively   
6 5 
  POLYNOMIALS
 
 
 
 
5) Verify that 2,-3 and 6 are the zeroes of the cubic polynomial 
 and then verify the relationship between the zeroes and the coefficients.
 Solution:  
 Compare the given polynomial with 
 a=1 b=-5 c=-12 and d=36. 
     Now to verify if 2,-3 and 6  are the roots of the polynomial.
     i.e. we have to show that 2	
     ? 
3 2
(2) 2 5(2) 12(2) 36 8 20 24 36 0 f = - - + = - - + =
 
3 2
( 3) ( 3) 5( 3) 12( 3) 36 27 45 36 36 0 f - = - - - - - + =- - + + =
 
3 2
(6) (6) 5(6) 12(6) 36 216 180 72 36 0 f = - - + = - - + =
 Therefore 2,-3 and 6 are the zeroes of the cubic polynomial 
 Let    =2 a  3 ß =- and 6 ? = 
  + 2 ( 3) 6 2 3 6 5 a ß ? + = + - + = - + = =
    2( 3) ( 3)6 6(2) 6 18 12 12 a a ß ß? ? + + = - + - + =- - + =- =
     2( 3)6 36
d
a
a
ß?
-
= - =- =  
 
6) If a and ß are the zeroes of the polynomial 
 Solution: 
 We know that  
(
(
Sum of zeroes=
Coefficient of x
-
               Product of the zeroes=
Coefficient of x
                   And hence,  
1 1
a ß aß
+ =
                                                     ?
 
 
 
 
POLYNOMIALS 
3 and 6 are the zeroes of the cubic polynomial 
3 2
( ) 5 12 36 f x x x x = - - +
elationship between the zeroes and the coefficients. 
Compare the given polynomial with 
3 2
ax bx cx d + + + we get, 
are the roots of the polynomial. 
 0 , 3 = 0 and 6  = 0 
(2) 2 5(2) 12(2) 36 8 20 24 36 0 = - - + = - - + = 
( 3) ( 3) 5( 3) 12( 3) 36 27 45 36 36 0 - = - - - - - + =- - + + = 
(6) (6) 5(6) 12(6) 36 216 180 72 36 0 = - - + = - - + = 
3 and 6 are the zeroes of the cubic polynomial 
3 2
5 12 36 x x x - - +
 + 2 ( 3) 6 2 3 6 5
b
a
-
+ = + - + = - + = = 
2( 3) ( 3)6 6(2) 6 18 12 12
c
a
+ + = - + - + =- - + =- = 
  
are the zeroes of the polynomial 
2
( ) f x ax bx c = + + then find 
1 1
a
+
( )
)
2
Coefficient of x
Coefficient of x
-
 
    i.e. 
b
a
a
ß
-
+ = 
2
Constant term
Coefficient of x
           i.e.   
c
a
a
ß = 
a ß
a ß aß
+
+ = 
b
a
c
a
-
=
b
c
-
= 
? 
1 1 b
c a ß
-
+ = 
( ) 5 12 36 f x x x x = - - + 
5 12 36  
1 1
ß
+ 
  POLYNOMIALS
 
 
 
 
7) Find the zeroes of the polynomial 
the coefficients 
 Solution: 
 Given that, 
 
2
( ) 4 5 4 3 5 f x x x = + - 
 
2
( ) 4 5 10 6 3 5 f x x x x = + - - 
 
( )( )
( ) 2 5 2 5 3 f x x x = + - 
 The zeroes of the polynomial are given by 
     i.e.         
( )( )
2 5 2 5 3 0 x x + - =
       ? 2 5 0 x+ = or 2 5 3 0 x- =
 
5
2
x
-
= or 
3
2 5
x= 
      Hence zeroes are 
3
2 5
a = and  
 a ß + = 
3 5 3 5 6 10 4 1
2 2 2 5 2 5 4 5 4 5 5
? ?
+ = - = = =
? ?
? ?
? ?
      
3 5 3
x
2 4 2 5
aß
? ?
- -
= =
? ?
? ?
? ?
 
      
4 1
4 5 5
b
a
- - -
= =   
3 5 3
4 5
c
a
- -
= =
      
b
a
a
ß
-
+ =    
c
a
a
ß = 
      
(
(
Coefficient of x
Sum of zeroes=   Product of the zeroes=
Coefficient of x
-
?
 
8) If a and ß are the zeroes of the polynomial 
Solution: 
We know that  
(
(
Sum of zeroes=
Coefficient of x
-
 
POLYNOMIALS
Find the zeroes of the polynomial 
2
( ) 4 5 4 3 5 f x x x = + - and verify the relationship between the zeroes and 
The zeroes of the polynomial are given by ( ) 0 f x = 
2 5 2 5 3 0  
2 5 3 0 - = 
and  
5
2
ß
-
= 
3 5 3 5 6 10 4 1
2 2 2 5 2 5 4 5 4 5 5
- - -
+ = - = = = 
3 5 3
4
- -
= = 
)
)
2
Coefficient of x
Constant term
Sum of zeroes=   Product of the zeroes=
Coefficient of x
Coefficient of x
are the zeroes of the polynomial 
2
( ) f x ax bx c = + +  then find 
a ß
ß a
+
( )
)
2
Coefficient of x
Coefficient of x
-
   i.e. 
b
a
a
ß
-
+ =
 
POLYNOMIALS 
verify the relationship between the zeroes and 
2
Constant term
Coefficient of x
 
a ß
ß a
+ 
Page 5


   POLYNOMIALS
 
 
 
GRAPHS OF POLYNOMIALS 
 
1) Draw a graph of the polynomial f x x
    Solution: 
 Let y = 3 2 x+ now we get different values of y corresponding to the different values of x. 
  
x 0 1 2 
 y 2 5 8 
 
 The points A(0,2), B(1,5) and C(2,8) are plotted on the graph paper on a suitable scale.
 The points are joint together to get the graph of the given polynomial. 
Since the polynomial is linear we get the graph as a straight line. 
 
          
 
RELATIONSHIP BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL
2) Find the zeroes of the quadratic polynomial 
     between the zeroes and the coefficients.
     Solution:  
     The given quadratic polynomial can be
     Thus the value of 
2
12 32 x x + + is zero when 
 i.e. when 8 x=- or 4 x=- Therefore the zeroes of 
 Sum of zeroes = -8+(-4)=-12 Product of the zeroes=
   
POLYNOMIALS 
( ) 3 2 f x x = + 
now we get different values of y corresponding to the different values of x. 
 
 
points A(0,2), B(1,5) and C(2,8) are plotted on the graph paper on a suitable scale.
The points are joint together to get the graph of the given polynomial.  
Since the polynomial is linear we get the graph as a straight line.  
BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL
Find the zeroes of the quadratic polynomial 
2
12 32 x x + + and verify the relationship 
between the zeroes and the coefficients. 
The given quadratic polynomial can be factorised as 
2
12 32 ( 8)( 4) x x x x + + = + +
is zero when ( 8) 0 x+ = or ( 4) 0 x+ = 
Therefore the zeroes of 
2
12 32 x x + + are -8 and -4. 
12 Product of the zeroes=-8 x -4=32 
now we get different values of y corresponding to the different values of x.  
points A(0,2), B(1,5) and C(2,8) are plotted on the graph paper on a suitable scale. 
 
BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL  
and verify the relationship  
12 32 ( 8)( 4) x x x x + + = + + 
4.  
                                       POLYNOMIALS
 
 
 
 
 
         
( )
( )
2
Coefficient of x
Coefficient of x
-
= 
(12)
1
-
= =-
  
(
(
Coefficient of x
Sum of zeroes=   Product of the zeroes=
Coefficient of x
-
?
 
3) Find the zeroes of the quadratic polynomial 
between the zeroes and the coefficients.
     Solution:      
Using the identity 
2 2
( )( ) a b a b a b - = + -
                  ?

v6

	
Hence, the value of 

v6

  is zero when 
Hence x = -v6 and x = v6 are the zeroes of 
Sum of zeroes 6 6 0 = - =  Product of zeroes
( )
( )
2
Coefficient of x
Coefficient of x
-
0
0
1
= = as the quadratic polynomial x
 
2
Constant term
Coefficient of x
6
6
1
-
= =- 
(
(
Coefficient of x
Sum of zeroes=   Product of the zeroes=
Coefficient of x
-
?
 
4) Find a quadratic polynomial, the sum and prod
 Solution; 
 Let the quadratic polynomial be ax bx c
 Given that  
   6 a ß + =-
b
a
-
= b a ? =
 and       x 5
c
a
a
ß = =    5 c a ? =
 If a=1, then b=6 and c=5  
 Thus, quadratic polynomial which satisfies the given condition is 
 
 
POLYNOMIALS 
(12)
12 = =-    
2
Constant term
Coefficient of x
 
32
32
1
= =
 
)
)
2
Coefficient of x
Constant term
Sum of zeroes=   Product of the zeroes=
Coefficient of x
Coefficient of x
Find the zeroes of the quadratic polynomial 
2
6 x - and verify the relationship  
between the zeroes and the coefficients. 
( )( ) a b a b a b - = + - we can write, 
v6		v6 
is zero when 6 0 x+ = or 6 0 x- = i.e. x=-
6 are the zeroes of the quadratic polynomial x
2
 - 6 
Product of zeroes 6 x - 6 6 = =- 
as the quadratic polynomial x
2
 – 6 we can write it as x
)
)
2
Coefficient of x
Constant term
Sum of zeroes=   Product of the zeroes=
Coefficient of x
Coefficient of x
Find a quadratic polynomial, the sum and product of whose zeroes are -6 and 5
2
ax bx c + + and its zeroes be a and ß 
6 b a ? = 
5 c a ? =  
quadratic polynomial which satisfies the given condition is 
2
6 5 x x + + 
2
Constant term
Coefficient of x
  
 
6 x=- or 6 x=
 
write it as x
2
 + 0x - 6 
2
Constant term
Coefficient of x
 
6 and 5 respectively   
6 5 
  POLYNOMIALS
 
 
 
 
5) Verify that 2,-3 and 6 are the zeroes of the cubic polynomial 
 and then verify the relationship between the zeroes and the coefficients.
 Solution:  
 Compare the given polynomial with 
 a=1 b=-5 c=-12 and d=36. 
     Now to verify if 2,-3 and 6  are the roots of the polynomial.
     i.e. we have to show that 2	
     ? 
3 2
(2) 2 5(2) 12(2) 36 8 20 24 36 0 f = - - + = - - + =
 
3 2
( 3) ( 3) 5( 3) 12( 3) 36 27 45 36 36 0 f - = - - - - - + =- - + + =
 
3 2
(6) (6) 5(6) 12(6) 36 216 180 72 36 0 f = - - + = - - + =
 Therefore 2,-3 and 6 are the zeroes of the cubic polynomial 
 Let    =2 a  3 ß =- and 6 ? = 
  + 2 ( 3) 6 2 3 6 5 a ß ? + = + - + = - + = =
    2( 3) ( 3)6 6(2) 6 18 12 12 a a ß ß? ? + + = - + - + =- - + =- =
     2( 3)6 36
d
a
a
ß?
-
= - =- =  
 
6) If a and ß are the zeroes of the polynomial 
 Solution: 
 We know that  
(
(
Sum of zeroes=
Coefficient of x
-
               Product of the zeroes=
Coefficient of x
                   And hence,  
1 1
a ß aß
+ =
                                                     ?
 
 
 
 
POLYNOMIALS 
3 and 6 are the zeroes of the cubic polynomial 
3 2
( ) 5 12 36 f x x x x = - - +
elationship between the zeroes and the coefficients. 
Compare the given polynomial with 
3 2
ax bx cx d + + + we get, 
are the roots of the polynomial. 
 0 , 3 = 0 and 6  = 0 
(2) 2 5(2) 12(2) 36 8 20 24 36 0 = - - + = - - + = 
( 3) ( 3) 5( 3) 12( 3) 36 27 45 36 36 0 - = - - - - - + =- - + + = 
(6) (6) 5(6) 12(6) 36 216 180 72 36 0 = - - + = - - + = 
3 and 6 are the zeroes of the cubic polynomial 
3 2
5 12 36 x x x - - +
 + 2 ( 3) 6 2 3 6 5
b
a
-
+ = + - + = - + = = 
2( 3) ( 3)6 6(2) 6 18 12 12
c
a
+ + = - + - + =- - + =- = 
  
are the zeroes of the polynomial 
2
( ) f x ax bx c = + + then find 
1 1
a
+
( )
)
2
Coefficient of x
Coefficient of x
-
 
    i.e. 
b
a
a
ß
-
+ = 
2
Constant term
Coefficient of x
           i.e.   
c
a
a
ß = 
a ß
a ß aß
+
+ = 
b
a
c
a
-
=
b
c
-
= 
? 
1 1 b
c a ß
-
+ = 
( ) 5 12 36 f x x x x = - - + 
5 12 36  
1 1
ß
+ 
  POLYNOMIALS
 
 
 
 
7) Find the zeroes of the polynomial 
the coefficients 
 Solution: 
 Given that, 
 
2
( ) 4 5 4 3 5 f x x x = + - 
 
2
( ) 4 5 10 6 3 5 f x x x x = + - - 
 
( )( )
( ) 2 5 2 5 3 f x x x = + - 
 The zeroes of the polynomial are given by 
     i.e.         
( )( )
2 5 2 5 3 0 x x + - =
       ? 2 5 0 x+ = or 2 5 3 0 x- =
 
5
2
x
-
= or 
3
2 5
x= 
      Hence zeroes are 
3
2 5
a = and  
 a ß + = 
3 5 3 5 6 10 4 1
2 2 2 5 2 5 4 5 4 5 5
? ?
+ = - = = =
? ?
? ?
? ?
      
3 5 3
x
2 4 2 5
aß
? ?
- -
= =
? ?
? ?
? ?
 
      
4 1
4 5 5
b
a
- - -
= =   
3 5 3
4 5
c
a
- -
= =
      
b
a
a
ß
-
+ =    
c
a
a
ß = 
      
(
(
Coefficient of x
Sum of zeroes=   Product of the zeroes=
Coefficient of x
-
?
 
8) If a and ß are the zeroes of the polynomial 
Solution: 
We know that  
(
(
Sum of zeroes=
Coefficient of x
-
 
POLYNOMIALS
Find the zeroes of the polynomial 
2
( ) 4 5 4 3 5 f x x x = + - and verify the relationship between the zeroes and 
The zeroes of the polynomial are given by ( ) 0 f x = 
2 5 2 5 3 0  
2 5 3 0 - = 
and  
5
2
ß
-
= 
3 5 3 5 6 10 4 1
2 2 2 5 2 5 4 5 4 5 5
- - -
+ = - = = = 
3 5 3
4
- -
= = 
)
)
2
Coefficient of x
Constant term
Sum of zeroes=   Product of the zeroes=
Coefficient of x
Coefficient of x
are the zeroes of the polynomial 
2
( ) f x ax bx c = + +  then find 
a ß
ß a
+
( )
)
2
Coefficient of x
Coefficient of x
-
   i.e. 
b
a
a
ß
-
+ =
 
POLYNOMIALS 
verify the relationship between the zeroes and 
2
Constant term
Coefficient of x
 
a ß
ß a
+ 
 POLYNOMIALS
 
 
 
 
Constant term
Product of the zeroes=
Coefficient of x
( )
2
2 2
2 a ß aß
a ß a ß
ß a aß aß
+ -
+
+ = =
?
2
2 b ac
ac
a ß
ß a
-
+ == 
9) If a and ß are the zeroes of the polynomial 
then find the value of k 
Solution: 
Since a and ß are zeroes of the polynomial.  
a +ß = 


 = -6,   aß = 


 = k   and a ß
We know that,   ( )
2 2
4 a ß aß a ß + - = -
Substituting the values we get,  
(-6)
2
 – 4k = 4
2      
 
36 – 4k = 16 
36 – 16 = 4k 
20 = 4k 
Therefore k = 5 
10)   Find the zeroes of the polynomial 
relationship between the zeroes and coefficients.
  Solution: 
  We have 
  
( )
2 2
2 2
( )
( )
( ) ( ) ( )
( ) ( )( )
f x abx b ac x bc
f x abx b x acx bc
f x bx ax b c ax b
f x ax b bx c
= + - -
= + - -
= + - +
= + -
 
POLYNOMIALS 
2
Constant term
Coefficient of x
  i.e.   
c
a
a
ß =
 
2 a ß aß
2
2 2
2 2
2
2
2
b c
b c b ac
a a
a a a
c c c
a a a
- ? ? ? ?
-
-
- ? ? ? ?
? ? ? ?
= = = =
es of the polynomial 
2
( ) 6 f x x x k = + + such that a ß - =
 are zeroes of the polynomial.   
4 a ß - = 
( )
2 2
a ß aß a ß + - = - 
Find the zeroes of the polynomial 
( )
2 2
( ) f x abx b ac x bc = + - - and verify the 
elationship between the zeroes and coefficients.  
f x abx b ac x bc
  
 
2
2 b ac
ac
-
= = = = 
4 a ß - =  
and verify the  
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