Class 10 Exam > Class 10 Notes > Mathematics (Maths) Class 10 > RD Sharma Solutions: Polynomials
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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
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