Question

if alpha and beta are zeroes of polynomial p(x)=x2+x+1 . Find 1÷alpha + 1÷beta

Answer 1

Hey there !!!!!

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p(x)=x²+x+1

A polynomial whose roots are α,β can be written as

   x²-(α+β)x+αβ=0

Comparing with x²+x+1   


          α+β = -1 and αβ = 1

1/α+1/β = α+β/αβ = -1/1 = -1


                 ( OR)


p(x)= x²+x+1

 α,β are roots of p(x)

p(x) will have imaginary roots as discriminant of p(x) < 0

       b²-4ac < 0


       1-4 < 0

So roots are complex.

p(x)=x²+x+1


x= (-b+√(b²-4ac))/2a


Here a=1 b=1 c= 1

x= -1+√1-4/2

x=-1+√-3/2

But √-3 is imaginary and in complex numbers i²=-1

So, √-3=√(3*i²)=√3i

x=-1+√3i/2 or -1-√3i/2

In complex numbers 

ω=-1+i√3/2          ω²=-1-i√3/2

So   ω,ω² are roots of p(x)=x²+x+1


  1/α + 1/β = α+β/αβ =  ω+ω²/ω*ω² 

In complex numbers ω³= 1 and ω+ω²=1

So    α+β/αβ = -1/1=-1

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Hope this helped you.......................



 

Answer 2

Answer:

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