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.

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 $\displaystyle \displaystyle \begin{align*} (z - z_1)(z - z_2)(z - z_3)(z - z_4)(z - z_5) \end{align*}$.