# Thread: Factoring 5th root polynomial

1. ## Factoring 5th root polynomial

My question started with
Show -243 in polar form which I did, <243, ∏ >
Then find all fifth roots of -243, which I did
z0 = <3, ∏/5 >
z1
. . . .
z4 = <3, 9∏/5 >

Now I need to factorise the polynomial z^5 + 243 giving exact values of the real coeff's. in terms of sin, cos where appropriate.

I think it should be simple for me, but need a quick start as to what is being looked for.

Thank you for looking.

2. ## Re: Factoring 5th root polynomial

Originally Posted by froodles01
My question started with
Show -243 in polar form which I did, <243, ∏ >
Then find all fifth roots of -243, which I did
z0 = <3, ∏/5 >
z1
. . . .
z4 = <3, 9∏/5 >

Now I need to factorise the polynomial z^5 + 243 giving exact values of the real coeff's. in terms of sin, cos where appropriate.

I think it should be simple for me, but need a quick start as to what is being looked for.

Thank you for looking.
Well you already have all the fifth roots, though the arguments should be in the region \displaystyle \begin{align*} \theta \in (-\pi, \pi] \end{align*}.

So the factorisation of \displaystyle \begin{align*} z^5 + 243 \end{align*} will be \displaystyle \begin{align*} (z - z_1)(z - z_2)(z - z_3)(z - z_4)(z - z_5) \end{align*}.