the equation x^2 +(1-2^2013)x + 2^2012(2^2012-1)-2=0 have roots a and b then value of (a-b)^2014=?
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Answer:
1
Step-by-step explanation:
Explanation:
Note that:
x
3
+
1
=
(
x
+
1
)
(
x
2
−
x
+
1
)
So since
α
is a root of
x
2
−
x
+
1
=
0
we have:
α
3
+
1
=
(
α
+
1
)
(
α
2
−
α
+
1
)
=
(
α
+
1
)
(
0
)
=
0
So:
α
3
=
−
1
Similarly:
β
3
=
−
1
Also:
x
2
−
x
+
1
=
(
x
−
α
)
(
x
−
β
)
=
x
2
−
(
α
+
β
)
x
+
α
β
So:
{
α
+
β
=
1
α
β
=
1
Note that:
α
2
+
β
2
=
(
α
+
β
)
2
−
2
α
β
=
1
−
2
=
−
1
So:
α
2009
+
β
2009
=
α
3
⋅
669
+
2
+
β
3
⋅
669
+
2
α
2009
+
β
2009
=
(
α
3
)
669
⋅
α
2
+
(
β
3
)
669
⋅
β
2
α
2009
+
β
2009
=
(
−
1
)
669
⋅
α
2
+
(
−
1
)
669
⋅
β
2
α
2009
+
β
2009
=
(
−
1
)
669
⋅
(
α
2
+
β
2
)
α
2009
+
β
2009
=
(
−
1
)
(
−
1
)
α
2009
+
β
2009
=
1
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