Hi, I've got a question which is the first part asks to show that (z4−eiθ)(z4−e−iθ)=z8−2z4cosθ+1 Which I've managed to do, however, the second part asked to hence solve the equation z8−z4−1=0 giving my answer in the form eiϕ where −π<ϕ≤π Though I recognise the similarities, I'm really not sure how to go about doing this Would somebody be able to help me? Thank you!
Hi, I've got a question which is the first part asks to show that (z4−eiθ)(z4−e−iθ)=z8−2z4cosθ+1 Which I've managed to do, however, the second part asked to hence solve the equation z8−z4−1=0 giving my answer in the form eiϕ where −π<ϕ≤π Though I recognise the similarities, I'm really not sure how to go about doing this Would somebody be able to help me? Thank you!
Hi, I've got a question which is the first part asks to show that (z4−eiθ)(z4−e−iθ)=z8−2z4cosθ+1 Which I've managed to do, however, the second part asked to hence solve the equation z8−z4−1=0 giving my answer in the form eiϕ where −π<ϕ≤π Though I recognise the similarities, I'm really not sure how to go about doing this Would somebody be able to help me? Thank you!
Here, theta is not related to the argument of the complex number z. The question labels the argument of the z variable using phi. Hence, theta is not restricted by the solutions to the polynomial.
Here, theta is not related to the argument of the complex number z. The question labels the argument of the z variable using phi. Hence, theta is not restricted by the solutions to the polynomial.
I understand. Would it be correct to say that the solutions we arrive at require the necessary condition that cosθ=0.5 to be met?
I.E the solution set is valid if and only if cosθ=0.5.
They don't really require that cos(θ)=0.5 rather, you are able to compare the two forms of the polynomials and notice that they agree when that condition is satisfied. Hence, you are able to write the latter polynomial in a more useful form.
They don't really require that cos(θ)=0.5 rather, you are able to compare the two forms of the polynomials and notice that they agree when that condition is satisfied. Hence, you are able to write the latter polynomial in a more useful form.
I think whats throwing me of is TeeMe wrote:
cosθ =1/2 θ =π /3
z4 = e^(iπ/4) and similarly the other
Shouldn't the third line be:
z4 = e^(iπ/3) and similarly the other
I.e, the working might be:
(z4−ei3π)(z4−e−i3π)=z8−z4+1
z4=ei3π,z4=e−i3π
Edit but z8−z4+1
Isn't the Octic we need to solve. So I am now confused again.