[Taylor_R]

z^2 - z +2 - (1/z) + (1/(z^2)) = 4 cos ^2 theta -2cos theta

hence solve the quartic equation

z^4 - z^3 +2(z^2) - z + 1 = 0

please help

[jj193]

can you solve the top equation in terms of cos theta? how does z relate to theta?

Notice how the two equations in z are related, can you make a nice factorisation?

[member910132]

(Original post by jj193)
can you solve the top equation in terms of cos theta? how does z relate to theta?

Notice how the two equations in z are related, can you make a nice factorisation?

I get stuck at I know that I can get thetha to be pi/2 or pi/3 but how do I proceed form there to get four values of z ?

[jj193]

you haven't solved for theta exhaustively if you want to consider this in polar co-ordinates with r>0

you should start slow
can zero be a solution? no
so you can express z as re^itheta r>0 zero in [0,2pi)
z=/=0 so has inverse
therefore

z^4 - z^3 +2(z^2) - z + 1 = 0 iff z^2 - z +2 - (1/z) + (1/(z^2)) =0

now in polar co-ordinates you know possible values of theta, can you figure out the values of r? (I honestly can't tell you how to do this atm) Maybe someone else will.

[BabyMaths]

(Original post by member910132)
I get stuck at I know that I can get thetha to be pi/2 or pi/3 but how do I proceed form there to get four values of z ?

Alternatively,





Factorise a little more and you're done.

Using the original approach you need to use the fact that so that

[member910132]

(Original post by BabyMaths)
Alternatively,





Factorise a little more and you're done.

Using the original approach you need to use the fact that so that

Using the second approach after geting

what do we do ? Solve for theta and what ?

Edit: I think I got it now, after getting theta = pi/2 and pi/3 we just use to get the values of z right ?

My answers are z = \pm i , -2 and 1

[member910132]

(Original post by jj193)
you haven't solved for theta exhaustively if you want to consider this in polar co-ordinates with r>0

you should start slow
can zero be a solution? no
so you can express z as re^itheta r>0 zero in [0,2pi)
z=/=0 so has inverse
therefore

z^4 - z^3 +2(z^2) - z + 1 = 0 iff z^2 - z +2 - (1/z) + (1/(z^2)) =0

now in polar co-ordinates you know possible values of theta, can you figure out the values of r? (I honestly can't tell you how to do this atm) Maybe someone else will.

Sorry I don't understand this in the slightest way at all

[member910132]

(Original post by Taylor_R)
z^2 - z +2 - (1/z) + (1/(z^2)) = 4 cos ^2 theta -2cos theta

hence solve the quartic equation

z^4 - z^3 +2(z^2) - z + 1 = 0

please help

What year is this from AQA right ?

[BabyMaths]

(Original post by member910132)
My answers are z = \pm i , -2 and 1

You don't need to ask if you have the correct values. Sub in 1 for example and you get 1-1+2-1+1=2.