Find all Zeros of a Polynomial Function given a Factor or Zero

We will use synthetic division to divide out a given zero or factor from a polynomial function. Then factor the quotient which is quadratic. As a bonus, I will sketch a rough graph of the polynomial function and write it in factored form! All good test questions to be prepared for!

[carultch]

I came up with a poem for the cubic formula:
x^3 + n*x = m
x = cbrt(m/2 + sqrt(D)) + cbrt(m/2 - sqrt(D))
D = (m/2)^2 + (n/3)^3

When x is cubed and x times n,
Are added and equal to m.
The values of x,
The goals of our quest,
Here's how to calculate them.

Cube roots to add,
Square roots they had,
Both of a term we'll call D.
Square half of m,
Cube third of n,
Add together and see.

Half of m, before the square root,
First we add with plus.
Its little brother,
Is just like the other,
Except with a sign of minus.

Cube rooting time, of both the brothers,
Add up the roots with glee.
We found our first x,
But where is the next?
I know there have to be three.

With help from DeMoivre,
Who's theorem, we love ya,
There's cube roots all over the plane
Yes, they're complex,
But do not perplex,
A new kind of numbers we gain.