RD Sharma Solutions: Polynomials

 Page 1


    
 
     Exercise 2.1 
 
1. Find the zeroes of each of the following quadratic polynomials and verify the relationship 
between the zeroes and their co efficient: 
(i) f(x) = ?? 2
- 2?? - 8  
(ii) g(s) = 4?? 2
- 4?? + 1 
(iii) h(t) = ?? 2
- 15 
(iv) p(x) = ?? 2
+ 2v2?? + 6 
(v) q(x) = v3?? 2
+ 10?? + 7v3 
(vi) f(x) = ?? 2
- ( v3 + 1) ?? + v3 
(vii) g(x) = ?? ( ?? 2
+ 1)- ?? ( ?? 2
+ 1) 
(viii) 6?? 2
- 3 - 7?? Sol: 
(i) f(x) = ?? 2
- 2?? - 8 
?? ( ?? ) = ?? 2
- 2?? - 8 = ?? 2
- 4?? + 2?? - 8  
= ?? ( ?? - 4)+ 2( ?? - 4) 
= ( ?? + 2) ( ?? - 4) 
Zeroes of the polynomials are -2 and 4 
Sum of the zeroes = 
- ???? ?????????????????? ???? ?? ???? ?????????????????? ???? ?? 
-2 + 4 = 
-( -2)
1
 
2 = 2 
Product of the zeroes = 
???????????????? ???????? ???? ?????????????????? ???? ?? 2
 
= 24 = 
-8
1
 
- 8 = -8  
? Hence the relationship verified 
(ii) 9( 5) = 45 - 45 + 1 = 45
2
- 25 - 25 + 1 = 25( 25 - 1)- 1( 25 - 1)  
= ( 25 - 1) ( 25 - 1) 
Zeroes of the polynomials are 
1
2
?????? 1
2
 
Sum of zeroes = 
- ???? ?????????????????? ???? ?? ???? ?????????????????? ???? ?? 2
 
1
2
+
1
2
=
-( -4)
4
  
1 = 1 
Product of the zeroes = 
???????????????? ???????? ???? ?????????????????? ???? ?? 2
 
1
2
×
1
2
=
1
4
?
1
4
=
1
4
  
? Hence the relationship verified. 
(iii) h(t) = ?? 2
- 15 = ( ?? 2
)- ( v15)
2
= ( ?? + v15) ( ?? - v15) 
zeroes of the polynomials are -v15 ?????? v15  
sum of zeroes = 0 
-v15 + v15 = 0  
0 = 0  
Page 2


    
 
     Exercise 2.1 
 
1. Find the zeroes of each of the following quadratic polynomials and verify the relationship 
between the zeroes and their co efficient: 
(i) f(x) = ?? 2
- 2?? - 8  
(ii) g(s) = 4?? 2
- 4?? + 1 
(iii) h(t) = ?? 2
- 15 
(iv) p(x) = ?? 2
+ 2v2?? + 6 
(v) q(x) = v3?? 2
+ 10?? + 7v3 
(vi) f(x) = ?? 2
- ( v3 + 1) ?? + v3 
(vii) g(x) = ?? ( ?? 2
+ 1)- ?? ( ?? 2
+ 1) 
(viii) 6?? 2
- 3 - 7?? Sol: 
(i) f(x) = ?? 2
- 2?? - 8 
?? ( ?? ) = ?? 2
- 2?? - 8 = ?? 2
- 4?? + 2?? - 8  
= ?? ( ?? - 4)+ 2( ?? - 4) 
= ( ?? + 2) ( ?? - 4) 
Zeroes of the polynomials are -2 and 4 
Sum of the zeroes = 
- ???? ?????????????????? ???? ?? ???? ?????????????????? ???? ?? 
-2 + 4 = 
-( -2)
1
 
2 = 2 
Product of the zeroes = 
???????????????? ???????? ???? ?????????????????? ???? ?? 2
 
= 24 = 
-8
1
 
- 8 = -8  
? Hence the relationship verified 
(ii) 9( 5) = 45 - 45 + 1 = 45
2
- 25 - 25 + 1 = 25( 25 - 1)- 1( 25 - 1)  
= ( 25 - 1) ( 25 - 1) 
Zeroes of the polynomials are 
1
2
?????? 1
2
 
Sum of zeroes = 
- ???? ?????????????????? ???? ?? ???? ?????????????????? ???? ?? 2
 
1
2
+
1
2
=
-( -4)
4
  
1 = 1 
Product of the zeroes = 
???????????????? ???????? ???? ?????????????????? ???? ?? 2
 
1
2
×
1
2
=
1
4
?
1
4
=
1
4
  
? Hence the relationship verified. 
(iii) h(t) = ?? 2
- 15 = ( ?? 2
)- ( v15)
2
= ( ?? + v15) ( ?? - v15) 
zeroes of the polynomials are -v15 ?????? v15  
sum of zeroes = 0 
-v15 + v15 = 0  
0 = 0  
    
 
Product of zeroes = 
-15
1
 
-v15 × v15 = -15  
-15 = -15 
? Hence the relationship verified. 
(iv) p(x) = ?? 2
+ 2v2?? - 6 = ?? 2
+ 3v2?? + v2 × 3v2 
= ?? ( ?? + 3v2)- v2( 2 + 3v2) = ( ?? - v2) ( ?? + 3v2) 
Zeroes of the polynomial are 3v2 and -3v2 
Sum of the zeroes = 
-3v2
1
 
v2 - 3v2 = -2v2  
-2v2 = -2v2  
?????????????? ???? ???????????? ? v2 × -3v2 = -
6
1
  
-6 = -6 
?????????? ?? h?? ???????????????? h???? ????????????????  
(v) 2(x) = v3?? 2
+ 10?? + 7v3 = v3?? 2
+ 7?? + 3?? + 7v3 
= v3?? ( ?? + v3)+ 7( ?? + v3) 
= ( v3?? + 7) ( ?? + v3) 
Zeroes of the polynomials are -v3,
-7
v3
 
Sum of zeroes = 
-10
v3
 
? -v3 -
7
v3
=
-10
v3
?
-10
v3
=
-10
v3
 
Product of zeroes = 
7v3
3
?
v3?? -7
v30
= 7 
? 7 = 7 
Hence, relationship verified. 
(vi) f(x) = ?? 2
- ( v3 + 1) ?? + v3 = ?? 2
- v3?? - ?? + v3 
= x (x - v3) – 1 (x - v3) 
= (x – 1) (x - v3)  
Zeroes of the polynomials are 1 and v3 
Sum of zeroes = 
-{?????????????????????? ???? ?? }
???? ?????????????????? ???? ?? 2
=
-[-v3-1]
1
 
1 + v3 = v3 + 1 
Product of zeroes = 
???????????????? ???????? ???? ?????????????????? ???? ?? 2
=
v3
1
 
1 × v3 = v3 = v3 = v3 
? Hence, relationship verified 
(vii) g(x) = ?? [( ?? 2
+ 1)- ?? ( ?? 2
+ 1) ]
2
= ?? ?? 2
+ ?? - ?? 2
?? - ?? 
= ?? ?? 2
- [( ?? 2
+ 1)- ?? ] + 0 = ?? ?? 2
- ?? 2
?? - ?? + ?? 
Page 3


    
 
     Exercise 2.1 
 
1. Find the zeroes of each of the following quadratic polynomials and verify the relationship 
between the zeroes and their co efficient: 
(i) f(x) = ?? 2
- 2?? - 8  
(ii) g(s) = 4?? 2
- 4?? + 1 
(iii) h(t) = ?? 2
- 15 
(iv) p(x) = ?? 2
+ 2v2?? + 6 
(v) q(x) = v3?? 2
+ 10?? + 7v3 
(vi) f(x) = ?? 2
- ( v3 + 1) ?? + v3 
(vii) g(x) = ?? ( ?? 2
+ 1)- ?? ( ?? 2
+ 1) 
(viii) 6?? 2
- 3 - 7?? Sol: 
(i) f(x) = ?? 2
- 2?? - 8 
?? ( ?? ) = ?? 2
- 2?? - 8 = ?? 2
- 4?? + 2?? - 8  
= ?? ( ?? - 4)+ 2( ?? - 4) 
= ( ?? + 2) ( ?? - 4) 
Zeroes of the polynomials are -2 and 4 
Sum of the zeroes = 
- ???? ?????????????????? ???? ?? ???? ?????????????????? ???? ?? 
-2 + 4 = 
-( -2)
1
 
2 = 2 
Product of the zeroes = 
???????????????? ???????? ???? ?????????????????? ???? ?? 2
 
= 24 = 
-8
1
 
- 8 = -8  
? Hence the relationship verified 
(ii) 9( 5) = 45 - 45 + 1 = 45
2
- 25 - 25 + 1 = 25( 25 - 1)- 1( 25 - 1)  
= ( 25 - 1) ( 25 - 1) 
Zeroes of the polynomials are 
1
2
?????? 1
2
 
Sum of zeroes = 
- ???? ?????????????????? ???? ?? ???? ?????????????????? ???? ?? 2
 
1
2
+
1
2
=
-( -4)
4
  
1 = 1 
Product of the zeroes = 
???????????????? ???????? ???? ?????????????????? ???? ?? 2
 
1
2
×
1
2
=
1
4
?
1
4
=
1
4
  
? Hence the relationship verified. 
(iii) h(t) = ?? 2
- 15 = ( ?? 2
)- ( v15)
2
= ( ?? + v15) ( ?? - v15) 
zeroes of the polynomials are -v15 ?????? v15  
sum of zeroes = 0 
-v15 + v15 = 0  
0 = 0  
    
 
Product of zeroes = 
-15
1
 
-v15 × v15 = -15  
-15 = -15 
? Hence the relationship verified. 
(iv) p(x) = ?? 2
+ 2v2?? - 6 = ?? 2
+ 3v2?? + v2 × 3v2 
= ?? ( ?? + 3v2)- v2( 2 + 3v2) = ( ?? - v2) ( ?? + 3v2) 
Zeroes of the polynomial are 3v2 and -3v2 
Sum of the zeroes = 
-3v2
1
 
v2 - 3v2 = -2v2  
-2v2 = -2v2  
?????????????? ???? ???????????? ? v2 × -3v2 = -
6
1
  
-6 = -6 
?????????? ?? h?? ???????????????? h???? ????????????????  
(v) 2(x) = v3?? 2
+ 10?? + 7v3 = v3?? 2
+ 7?? + 3?? + 7v3 
= v3?? ( ?? + v3)+ 7( ?? + v3) 
= ( v3?? + 7) ( ?? + v3) 
Zeroes of the polynomials are -v3,
-7
v3
 
Sum of zeroes = 
-10
v3
 
? -v3 -
7
v3
=
-10
v3
?
-10
v3
=
-10
v3
 
Product of zeroes = 
7v3
3
?
v3?? -7
v30
= 7 
? 7 = 7 
Hence, relationship verified. 
(vi) f(x) = ?? 2
- ( v3 + 1) ?? + v3 = ?? 2
- v3?? - ?? + v3 
= x (x - v3) – 1 (x - v3) 
= (x – 1) (x - v3)  
Zeroes of the polynomials are 1 and v3 
Sum of zeroes = 
-{?????????????????????? ???? ?? }
???? ?????????????????? ???? ?? 2
=
-[-v3-1]
1
 
1 + v3 = v3 + 1 
Product of zeroes = 
???????????????? ???????? ???? ?????????????????? ???? ?? 2
=
v3
1
 
1 × v3 = v3 = v3 = v3 
? Hence, relationship verified 
(vii) g(x) = ?? [( ?? 2
+ 1)- ?? ( ?? 2
+ 1) ]
2
= ?? ?? 2
+ ?? - ?? 2
?? - ?? 
= ?? ?? 2
- [( ?? 2
+ 1)- ?? ] + 0 = ?? ?? 2
- ?? 2
?? - ?? + ?? 
    
 
= ???? ( ?? - ?? )- 1( ?? - ?? ) = ( ?? - ?? ) ( ???? - 1) 
Zeroes of the polynomials = 
1
?? ?????? ??  
Sum of the zeroes = 
-[-?? 2
-1]
?? 
? 
1
?? + ?? =
?? 2
+1
?? ?
?? 2
+1
?? =
?? 2
+1
?? 
Product of zeroes = 
?? ?? 
? 
1
?? × ?? =
?? ?? ?
?? 2
+1
?? =
?? 2
+1
?? 
Product of zeroes = 
?? ?? ? 1 = 1 
Hence relationship verified 
(viii) 6?? 2
- 3 - 7?? = 6?? 2
- 7?? - 3 = ( 3?? + 11) ( 2?? - 3) 
Zeroes of polynomials are +
3
2
?????? -1
3
  
Sum of zeroes = 
-1
3
+
3
2
=
7
6
=
-( -7)
6
=
-( ???? ?????????????????? ???? ?? )
???? ?????????????????? ???? ?? 2
 
Product of zeroes = 
-1
3
×
3
2
=
-1
2
=
-3
6
=
???????????????? ???????? ???? ?????????????????? ???? ?? 2
 
? Hence, relationship verified. 
 
2. If ?? and ?? are the zeros of the quadratic polynomial f(x) = ax
2
 + bx + c, then evaluate: 
(i) ?? - ?? 
(ii) 
1
?? -
1
??  
(iii) 
1
?? +
1
?? - 2?? ?? 
(iv) ?? 2
 ?? + ?? ?? 2
 
(v) ?? 4
+ ?? 4
 
(vi) 
1
???? +?? +
1
???? +?? 
(vii) 
?? ???? +?? +
?? ???? +?? 
(viii) ?? [
?? 2
?? +
?? 2
?? ] +
?? [
?? ?? +
?? ?? ] 
Sol: 
f(x) = ?? ?? 2
+ ???? + ?? 
?? + ?? =
-?? ??  
???? =
?? ??  
?????????? ?? + ?? ?????? ?? h?? ?????????? ( ???? ) ???????????? ???? ?? h?? ?????????? ??????????????????????   
(i) ?? - ?? 
The two zeroes of the polynomials are 
-?? +v?? 2
-4????
2?? - (?? -v?? 2
-4????
2?? ) = -?? +
v?? 2
-4???? +?? +v?? 2
-4????
2?? =
2v?? 2
-4????
2?? =
v?? 2
-4????
2??  
(ii) 
1
?? -
1
?? =
?? -?? ????
=
-( ?? - ?? )
????
… ( ?? ) 
From (i) we know that ?? - ?? =
v?? 2
-4????
2?? [???????? ( ?? ) ]???? =
?? ?? 
Putting the values in the (a) = - (
v?? 2
-4???? ×?? ?? ×?? ) =
-v?? 2
-4????
?? 
(iii) 
1
?? +
1
?? - 2?? ?? 
Page 4


    
 
     Exercise 2.1 
 
1. Find the zeroes of each of the following quadratic polynomials and verify the relationship 
between the zeroes and their co efficient: 
(i) f(x) = ?? 2
- 2?? - 8  
(ii) g(s) = 4?? 2
- 4?? + 1 
(iii) h(t) = ?? 2
- 15 
(iv) p(x) = ?? 2
+ 2v2?? + 6 
(v) q(x) = v3?? 2
+ 10?? + 7v3 
(vi) f(x) = ?? 2
- ( v3 + 1) ?? + v3 
(vii) g(x) = ?? ( ?? 2
+ 1)- ?? ( ?? 2
+ 1) 
(viii) 6?? 2
- 3 - 7?? Sol: 
(i) f(x) = ?? 2
- 2?? - 8 
?? ( ?? ) = ?? 2
- 2?? - 8 = ?? 2
- 4?? + 2?? - 8  
= ?? ( ?? - 4)+ 2( ?? - 4) 
= ( ?? + 2) ( ?? - 4) 
Zeroes of the polynomials are -2 and 4 
Sum of the zeroes = 
- ???? ?????????????????? ???? ?? ???? ?????????????????? ???? ?? 
-2 + 4 = 
-( -2)
1
 
2 = 2 
Product of the zeroes = 
???????????????? ???????? ???? ?????????????????? ???? ?? 2
 
= 24 = 
-8
1
 
- 8 = -8  
? Hence the relationship verified 
(ii) 9( 5) = 45 - 45 + 1 = 45
2
- 25 - 25 + 1 = 25( 25 - 1)- 1( 25 - 1)  
= ( 25 - 1) ( 25 - 1) 
Zeroes of the polynomials are 
1
2
?????? 1
2
 
Sum of zeroes = 
- ???? ?????????????????? ???? ?? ???? ?????????????????? ???? ?? 2
 
1
2
+
1
2
=
-( -4)
4
  
1 = 1 
Product of the zeroes = 
???????????????? ???????? ???? ?????????????????? ???? ?? 2
 
1
2
×
1
2
=
1
4
?
1
4
=
1
4
  
? Hence the relationship verified. 
(iii) h(t) = ?? 2
- 15 = ( ?? 2
)- ( v15)
2
= ( ?? + v15) ( ?? - v15) 
zeroes of the polynomials are -v15 ?????? v15  
sum of zeroes = 0 
-v15 + v15 = 0  
0 = 0  
    
 
Product of zeroes = 
-15
1
 
-v15 × v15 = -15  
-15 = -15 
? Hence the relationship verified. 
(iv) p(x) = ?? 2
+ 2v2?? - 6 = ?? 2
+ 3v2?? + v2 × 3v2 
= ?? ( ?? + 3v2)- v2( 2 + 3v2) = ( ?? - v2) ( ?? + 3v2) 
Zeroes of the polynomial are 3v2 and -3v2 
Sum of the zeroes = 
-3v2
1
 
v2 - 3v2 = -2v2  
-2v2 = -2v2  
?????????????? ???? ???????????? ? v2 × -3v2 = -
6
1
  
-6 = -6 
?????????? ?? h?? ???????????????? h???? ????????????????  
(v) 2(x) = v3?? 2
+ 10?? + 7v3 = v3?? 2
+ 7?? + 3?? + 7v3 
= v3?? ( ?? + v3)+ 7( ?? + v3) 
= ( v3?? + 7) ( ?? + v3) 
Zeroes of the polynomials are -v3,
-7
v3
 
Sum of zeroes = 
-10
v3
 
? -v3 -
7
v3
=
-10
v3
?
-10
v3
=
-10
v3
 
Product of zeroes = 
7v3
3
?
v3?? -7
v30
= 7 
? 7 = 7 
Hence, relationship verified. 
(vi) f(x) = ?? 2
- ( v3 + 1) ?? + v3 = ?? 2
- v3?? - ?? + v3 
= x (x - v3) – 1 (x - v3) 
= (x – 1) (x - v3)  
Zeroes of the polynomials are 1 and v3 
Sum of zeroes = 
-{?????????????????????? ???? ?? }
???? ?????????????????? ???? ?? 2
=
-[-v3-1]
1
 
1 + v3 = v3 + 1 
Product of zeroes = 
???????????????? ???????? ???? ?????????????????? ???? ?? 2
=
v3
1
 
1 × v3 = v3 = v3 = v3 
? Hence, relationship verified 
(vii) g(x) = ?? [( ?? 2
+ 1)- ?? ( ?? 2
+ 1) ]
2
= ?? ?? 2
+ ?? - ?? 2
?? - ?? 
= ?? ?? 2
- [( ?? 2
+ 1)- ?? ] + 0 = ?? ?? 2
- ?? 2
?? - ?? + ?? 
    
 
= ???? ( ?? - ?? )- 1( ?? - ?? ) = ( ?? - ?? ) ( ???? - 1) 
Zeroes of the polynomials = 
1
?? ?????? ??  
Sum of the zeroes = 
-[-?? 2
-1]
?? 
? 
1
?? + ?? =
?? 2
+1
?? ?
?? 2
+1
?? =
?? 2
+1
?? 
Product of zeroes = 
?? ?? 
? 
1
?? × ?? =
?? ?? ?
?? 2
+1
?? =
?? 2
+1
?? 
Product of zeroes = 
?? ?? ? 1 = 1 
Hence relationship verified 
(viii) 6?? 2
- 3 - 7?? = 6?? 2
- 7?? - 3 = ( 3?? + 11) ( 2?? - 3) 
Zeroes of polynomials are +
3
2
?????? -1
3
  
Sum of zeroes = 
-1
3
+
3
2
=
7
6
=
-( -7)
6
=
-( ???? ?????????????????? ???? ?? )
???? ?????????????????? ???? ?? 2
 
Product of zeroes = 
-1
3
×
3
2
=
-1
2
=
-3
6
=
???????????????? ???????? ???? ?????????????????? ???? ?? 2
 
? Hence, relationship verified. 
 
2. If ?? and ?? are the zeros of the quadratic polynomial f(x) = ax
2
 + bx + c, then evaluate: 
(i) ?? - ?? 
(ii) 
1
?? -
1
??  
(iii) 
1
?? +
1
?? - 2?? ?? 
(iv) ?? 2
 ?? + ?? ?? 2
 
(v) ?? 4
+ ?? 4
 
(vi) 
1
???? +?? +
1
???? +?? 
(vii) 
?? ???? +?? +
?? ???? +?? 
(viii) ?? [
?? 2
?? +
?? 2
?? ] +
?? [
?? ?? +
?? ?? ] 
Sol: 
f(x) = ?? ?? 2
+ ???? + ?? 
?? + ?? =
-?? ??  
???? =
?? ??  
?????????? ?? + ?? ?????? ?? h?? ?????????? ( ???? ) ???????????? ???? ?? h?? ?????????? ??????????????????????   
(i) ?? - ?? 
The two zeroes of the polynomials are 
-?? +v?? 2
-4????
2?? - (?? -v?? 2
-4????
2?? ) = -?? +
v?? 2
-4???? +?? +v?? 2
-4????
2?? =
2v?? 2
-4????
2?? =
v?? 2
-4????
2??  
(ii) 
1
?? -
1
?? =
?? -?? ????
=
-( ?? - ?? )
????
… ( ?? ) 
From (i) we know that ?? - ?? =
v?? 2
-4????
2?? [???????? ( ?? ) ]???? =
?? ?? 
Putting the values in the (a) = - (
v?? 2
-4???? ×?? ?? ×?? ) =
-v?? 2
-4????
?? 
(iii) 
1
?? +
1
?? - 2?? ?? 
    
 
? [
?? + ?? ????
] - 2????    
?
-?? ?? ×
?? ?? - 2
?? ?? = -2
?? ?? -
?? ?? =
-???? -2?? 2
????
- [
?? ?? +
2?? ?? ]   
(iv) ?? 2
 ?? + ?? ?? 2
 
?? ?? ( ?? + ?? )    
= 
?? ?? (
-?? ?? ) 
= 
-????
?? 2
 
(v) ?? 4
+ ?? 4
= ( ?? 2
+ ?? 2
)
2
- 2?? 2
+ ?? 2
 
= ( ( ?? + ?? )
2
- 2???? )
2
- 2( ???? )
2
  
= [(-
?? ?? )
2
- 2
?? ?? ]
2
- [2 (
?? ?? )
2
]  
= [
?? 2
-2????
?? 2
]
2
-
2?? 2
?? 2
    
=
( ?? 2
2???? )
2
-2?? 2
?? 2
?? 4
  
(vi) 
1
???? +?? +
1
???? +?? 
?
???? +?? +???? +?? ( 3?? +?? ) ( ???? +?? )
  
=
?? ( ?? + ?? ) +2?? ?? 2
???? +???? ?? +?????? +?? 2
  
=
?? ( ?? +?? ) +?? ?? 2
???? +???? ( ?? 2
?? ) +?? 2
  
=
?? ×
?? +2?? ?? ?? ×
?? ?? +
?????? ( -?? ) +?? 2
?? =
?? ???? -?? 2
+?? 2
=
?? ????
  
(vii) 
?? ???? +?? +
?? ???? +?? 
= 
?? ( ???? +?? ) +?? ( ???? +?? )
( ???? +?? ) ( ???? +?? )
 
= 
?? ?? 2
+???? +?? ?? 2
+????
?? 2
???? +?????? +?????? +?? 2
 
= 
?? ?? 2
+?? ?? 2
+?? ?? 2
+????
?? ×
?? ?? +???? ( ?? +?? ) +?? 2
 
=
?? [( ?? 2
+?? 2
) +?? ( ?? +?? ) ]
???? +???? +?? (
-?? ?? )+?? 2
  
=
?? [( ?? + ?? )
2
-2???? ]+???? -
?? ?? ????
  
=
?? [
?? 2
?? -
2?? ?? ]-
?? 2
?? ????
=
?? ×[
?? 2
-2?? ?? ]-?? 2
????
=
-2
??  
(viii) ?? [
?? 2
?? +
?? 2
?? ] + ?? [
?? ?? +
?? ?? ] 
= ?? [
?? 3
+?? 3
????
] + ?? (
?? 2
+?? 2
????
)  
Page 5


    
 
     Exercise 2.1 
 
1. Find the zeroes of each of the following quadratic polynomials and verify the relationship 
between the zeroes and their co efficient: 
(i) f(x) = ?? 2
- 2?? - 8  
(ii) g(s) = 4?? 2
- 4?? + 1 
(iii) h(t) = ?? 2
- 15 
(iv) p(x) = ?? 2
+ 2v2?? + 6 
(v) q(x) = v3?? 2
+ 10?? + 7v3 
(vi) f(x) = ?? 2
- ( v3 + 1) ?? + v3 
(vii) g(x) = ?? ( ?? 2
+ 1)- ?? ( ?? 2
+ 1) 
(viii) 6?? 2
- 3 - 7?? Sol: 
(i) f(x) = ?? 2
- 2?? - 8 
?? ( ?? ) = ?? 2
- 2?? - 8 = ?? 2
- 4?? + 2?? - 8  
= ?? ( ?? - 4)+ 2( ?? - 4) 
= ( ?? + 2) ( ?? - 4) 
Zeroes of the polynomials are -2 and 4 
Sum of the zeroes = 
- ???? ?????????????????? ???? ?? ???? ?????????????????? ???? ?? 
-2 + 4 = 
-( -2)
1
 
2 = 2 
Product of the zeroes = 
???????????????? ???????? ???? ?????????????????? ???? ?? 2
 
= 24 = 
-8
1
 
- 8 = -8  
? Hence the relationship verified 
(ii) 9( 5) = 45 - 45 + 1 = 45
2
- 25 - 25 + 1 = 25( 25 - 1)- 1( 25 - 1)  
= ( 25 - 1) ( 25 - 1) 
Zeroes of the polynomials are 
1
2
?????? 1
2
 
Sum of zeroes = 
- ???? ?????????????????? ???? ?? ???? ?????????????????? ???? ?? 2
 
1
2
+
1
2
=
-( -4)
4
  
1 = 1 
Product of the zeroes = 
???????????????? ???????? ???? ?????????????????? ???? ?? 2
 
1
2
×
1
2
=
1
4
?
1
4
=
1
4
  
? Hence the relationship verified. 
(iii) h(t) = ?? 2
- 15 = ( ?? 2
)- ( v15)
2
= ( ?? + v15) ( ?? - v15) 
zeroes of the polynomials are -v15 ?????? v15  
sum of zeroes = 0 
-v15 + v15 = 0  
0 = 0  
    
 
Product of zeroes = 
-15
1
 
-v15 × v15 = -15  
-15 = -15 
? Hence the relationship verified. 
(iv) p(x) = ?? 2
+ 2v2?? - 6 = ?? 2
+ 3v2?? + v2 × 3v2 
= ?? ( ?? + 3v2)- v2( 2 + 3v2) = ( ?? - v2) ( ?? + 3v2) 
Zeroes of the polynomial are 3v2 and -3v2 
Sum of the zeroes = 
-3v2
1
 
v2 - 3v2 = -2v2  
-2v2 = -2v2  
?????????????? ???? ???????????? ? v2 × -3v2 = -
6
1
  
-6 = -6 
?????????? ?? h?? ???????????????? h???? ????????????????  
(v) 2(x) = v3?? 2
+ 10?? + 7v3 = v3?? 2
+ 7?? + 3?? + 7v3 
= v3?? ( ?? + v3)+ 7( ?? + v3) 
= ( v3?? + 7) ( ?? + v3) 
Zeroes of the polynomials are -v3,
-7
v3
 
Sum of zeroes = 
-10
v3
 
? -v3 -
7
v3
=
-10
v3
?
-10
v3
=
-10
v3
 
Product of zeroes = 
7v3
3
?
v3?? -7
v30
= 7 
? 7 = 7 
Hence, relationship verified. 
(vi) f(x) = ?? 2
- ( v3 + 1) ?? + v3 = ?? 2
- v3?? - ?? + v3 
= x (x - v3) – 1 (x - v3) 
= (x – 1) (x - v3)  
Zeroes of the polynomials are 1 and v3 
Sum of zeroes = 
-{?????????????????????? ???? ?? }
???? ?????????????????? ???? ?? 2
=
-[-v3-1]
1
 
1 + v3 = v3 + 1 
Product of zeroes = 
???????????????? ???????? ???? ?????????????????? ???? ?? 2
=
v3
1
 
1 × v3 = v3 = v3 = v3 
? Hence, relationship verified 
(vii) g(x) = ?? [( ?? 2
+ 1)- ?? ( ?? 2
+ 1) ]
2
= ?? ?? 2
+ ?? - ?? 2
?? - ?? 
= ?? ?? 2
- [( ?? 2
+ 1)- ?? ] + 0 = ?? ?? 2
- ?? 2
?? - ?? + ?? 
    
 
= ???? ( ?? - ?? )- 1( ?? - ?? ) = ( ?? - ?? ) ( ???? - 1) 
Zeroes of the polynomials = 
1
?? ?????? ??  
Sum of the zeroes = 
-[-?? 2
-1]
?? 
? 
1
?? + ?? =
?? 2
+1
?? ?
?? 2
+1
?? =
?? 2
+1
?? 
Product of zeroes = 
?? ?? 
? 
1
?? × ?? =
?? ?? ?
?? 2
+1
?? =
?? 2
+1
?? 
Product of zeroes = 
?? ?? ? 1 = 1 
Hence relationship verified 
(viii) 6?? 2
- 3 - 7?? = 6?? 2
- 7?? - 3 = ( 3?? + 11) ( 2?? - 3) 
Zeroes of polynomials are +
3
2
?????? -1
3
  
Sum of zeroes = 
-1
3
+
3
2
=
7
6
=
-( -7)
6
=
-( ???? ?????????????????? ???? ?? )
???? ?????????????????? ???? ?? 2
 
Product of zeroes = 
-1
3
×
3
2
=
-1
2
=
-3
6
=
???????????????? ???????? ???? ?????????????????? ???? ?? 2
 
? Hence, relationship verified. 
 
2. If ?? and ?? are the zeros of the quadratic polynomial f(x) = ax
2
 + bx + c, then evaluate: 
(i) ?? - ?? 
(ii) 
1
?? -
1
??  
(iii) 
1
?? +
1
?? - 2?? ?? 
(iv) ?? 2
 ?? + ?? ?? 2
 
(v) ?? 4
+ ?? 4
 
(vi) 
1
???? +?? +
1
???? +?? 
(vii) 
?? ???? +?? +
?? ???? +?? 
(viii) ?? [
?? 2
?? +
?? 2
?? ] +
?? [
?? ?? +
?? ?? ] 
Sol: 
f(x) = ?? ?? 2
+ ???? + ?? 
?? + ?? =
-?? ??  
???? =
?? ??  
?????????? ?? + ?? ?????? ?? h?? ?????????? ( ???? ) ???????????? ???? ?? h?? ?????????? ??????????????????????   
(i) ?? - ?? 
The two zeroes of the polynomials are 
-?? +v?? 2
-4????
2?? - (?? -v?? 2
-4????
2?? ) = -?? +
v?? 2
-4???? +?? +v?? 2
-4????
2?? =
2v?? 2
-4????
2?? =
v?? 2
-4????
2??  
(ii) 
1
?? -
1
?? =
?? -?? ????
=
-( ?? - ?? )
????
… ( ?? ) 
From (i) we know that ?? - ?? =
v?? 2
-4????
2?? [???????? ( ?? ) ]???? =
?? ?? 
Putting the values in the (a) = - (
v?? 2
-4???? ×?? ?? ×?? ) =
-v?? 2
-4????
?? 
(iii) 
1
?? +
1
?? - 2?? ?? 
    
 
? [
?? + ?? ????
] - 2????    
?
-?? ?? ×
?? ?? - 2
?? ?? = -2
?? ?? -
?? ?? =
-???? -2?? 2
????
- [
?? ?? +
2?? ?? ]   
(iv) ?? 2
 ?? + ?? ?? 2
 
?? ?? ( ?? + ?? )    
= 
?? ?? (
-?? ?? ) 
= 
-????
?? 2
 
(v) ?? 4
+ ?? 4
= ( ?? 2
+ ?? 2
)
2
- 2?? 2
+ ?? 2
 
= ( ( ?? + ?? )
2
- 2???? )
2
- 2( ???? )
2
  
= [(-
?? ?? )
2
- 2
?? ?? ]
2
- [2 (
?? ?? )
2
]  
= [
?? 2
-2????
?? 2
]
2
-
2?? 2
?? 2
    
=
( ?? 2
2???? )
2
-2?? 2
?? 2
?? 4
  
(vi) 
1
???? +?? +
1
???? +?? 
?
???? +?? +???? +?? ( 3?? +?? ) ( ???? +?? )
  
=
?? ( ?? + ?? ) +2?? ?? 2
???? +???? ?? +?????? +?? 2
  
=
?? ( ?? +?? ) +?? ?? 2
???? +???? ( ?? 2
?? ) +?? 2
  
=
?? ×
?? +2?? ?? ?? ×
?? ?? +
?????? ( -?? ) +?? 2
?? =
?? ???? -?? 2
+?? 2
=
?? ????
  
(vii) 
?? ???? +?? +
?? ???? +?? 
= 
?? ( ???? +?? ) +?? ( ???? +?? )
( ???? +?? ) ( ???? +?? )
 
= 
?? ?? 2
+???? +?? ?? 2
+????
?? 2
???? +?????? +?????? +?? 2
 
= 
?? ?? 2
+?? ?? 2
+?? ?? 2
+????
?? ×
?? ?? +???? ( ?? +?? ) +?? 2
 
=
?? [( ?? 2
+?? 2
) +?? ( ?? +?? ) ]
???? +???? +?? (
-?? ?? )+?? 2
  
=
?? [( ?? + ?? )
2
-2???? ]+???? -
?? ?? ????
  
=
?? [
?? 2
?? -
2?? ?? ]-
?? 2
?? ????
=
?? ×[
?? 2
-2?? ?? ]-?? 2
????
=
-2
??  
(viii) ?? [
?? 2
?? +
?? 2
?? ] + ?? [
?? ?? +
?? ?? ] 
= ?? [
?? 3
+?? 3
????
] + ?? (
?? 2
+?? 2
????
)  
    
 
=
?? [( ?? +?? )
3
-3???? ( ?? +?? ) ]
?? ?? + ?? ( ?? + ?? )
2
- 2?? ??  
=
?? [(
-?? 3
?? 3
)+
3?? ?? .
?? ?? +?? (
?? 2
?? 2
-
2?? ?? )]
?? ??  
= 
?? 2
?? [
-?? 3
?? 3
+
3????
?? 2
+
?? 3
?? 2
-
2????
?? ]   
= 
-?? 2
?? 3
?? ?? 3
+
3?? 2
????
?? ?? 2
+
?? 3
?? 2
?? 2
?? -
2???? ?? 2
????
 
= 
-?? 3
????
+ 3?? +
?? 3
????
- 2?? 
= b 
 
3. If ?? and ?? are the zeros of the quadratic polynomial f(x) = 6x
2
 + x - 2, find the value of 
?? ?? +
?? ??  
Sol: 
f(x) = 6?? 2
- ?? - 2 
Since ?? and ?? are the zeroes of the given polynomial  
? Sum of zeroes [?? + ?? ] = 
-1
6
 
Product of zeroes (???? ) = 
-1
3
 
= 
?? ?? +
?? ?? =
?? 2
+?? 2
????
=
( ?? + ?? )
2
-2????
????
 
= 
(
1
6
)
2
-2×(
-1
3
)
-
1
3
=
1
6
-
2
3
-1
3
=
1+24
36
-1
3
 
= 
2
36
1
3
=
-25
12
 
 
4. If a and  are the zeros of the quadratic polynomial f(x) = ?? 2
- ?? - 4, find the value of  
1
?? +
1
?? - ????  
Sol: 
Since ?? + ?? are the zeroes of the polynomial: ?? 2
- ?? - 4 
Sum of the roots (?? + ?? ) = 1 
Product of the roots (???? ) = -4 
1
?? +
1
?? - ???? =
?? +?? ????
- ????  
=
1
-4
+ 4 =
-1
4
+ 4 =
-1+16
4
=
15
4
  
 
5. If ?? and ?? are the zeros of the quadratic polynomial p(x) = 4x
2
 - 5x -1, find the value of  
?? 2
?? + ?? ?? 2
. 
Sol: 
Since ?? ?????? ?? are the roots of the polynomial: 4?? 2
- 5?? - 1