the common roots of equations z³+2z²+2z+1=0 and z²⁰¹⁴+z²⁰¹⁵+1=0 are
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Solution -
z
3
+2z
2
+2z+1=0 ___ (1)
z
1985
+z
100
+1=0 ___ (2)
We know that 1+ω+ω
2
=0,ω
3
=1
check to be root,
(1) ω
3
+2ω
2
+2ω+1
=1+ω+ω
2
+1+ω+ω
2
=0+0=0
(2) ω
1985
+ω
100
+1
ω
2
+ω+1=0
ω is a common root
check ω
2
to be a root
(1) ω
6
+2ω
4
+2ω
2
+1
=(ω
3
)
2
+2ω(ω
3
)+2ω
2
+1
=1+ω+ω
2
+1+ω+ω
2
=0
(2) =ω
3970
+ω
200
+1
=ω+ω
2
+1=0
ω
2 is a common root
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