I need help with complex number

could anyone explain how to do this question?

Prove that 1+i is a root of the equation z^4 + 3z^2 - 6z + 10 = 0
Find all the other roots

Thanks

Reply 1

freemyice

just substitue z=1+i into f(z)=z^4 + 3z^2 - 6z + 10
if f(1+i)=0, then 1+i is a root.

Reply 2

Aitch

2

rlagksquf

could anyone explain how to do this question?

Prove that 1+i is a root of the equation z^4 + 3z^2 - 6z + 10 = 0

Thanks

If 1+i is a root, then 1-i is a root.
so (z-1-i) and (z-1+i) are factors.
so (z-1-i)(z-1+i) is a factor, = z²-2z+2

Long division (remember to add 0z³ before you start) gives the other factor as z² + 2z + 5

Aitch

Reply 3

rlagksquf

OP

Oops..sorry

I didn't put "Find all the other roots" :>.<:

my answer says "1-i, -1 + 2i , -1 - 2i

Reply 4

Aitch

2

rlagksquf

Oops..sorry

I didn't put "Find all the other roots" :>.<:

my answer says "1-i, -1 + 2i , -1 - 2i

Take z²+2z+5 from the end of my previous post, and put the coefficients into the quadratic formula.

Get z= (-2±√(4-20))/2

Solve to get the other 2 roots

Aitch

Reply 5

rlagksquf

OP

Aitch

If 1+i is a root, then 1-i is a root.
so (z-1-i) and (z-1+i) are factors.
so (z-1-i)(z-1+i) is a factor, = z²-2z+2

Long division (remember to add 0z³ before you start) gives the other factor as z² + 2z + 5

Aitch

thanks

I need help with complex number

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