# Factoring 5th root polynomial

• May 2nd 2012, 02:15 AM
froodles01
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.
• May 2nd 2012, 06:01 AM
Prove It
Re: Factoring 5th root polynomial
Quote:

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 \displaystyle \begin{align*} \theta \in (-\pi, \pi] \end{align*}.

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