# Math Help - Polynomial equation

1. ## Polynomial equation

Determine the polynomial equation in standard form that has the following roots:

-1 (of order 2), and (2+2sqrt 3) and (2-2sqrt 3)

please show all work!

2. Hello, meli3000!

Determine the polynomial equation in standard form that has the following roots:

. . $-1\:\text{(of order 2)},\;\,2+2\sqrt{3},\;\;2-2\sqrt{3}$

If the polynomial $P(x)$ has root $-1\text{ (order 2)}$, then $(x+1)^2$ is a factor of $P(x)$.

If $2 + 2\sqrt{3}$ is a root, then $x - (2 + 2\sqrt{3})$ is a factor.

If $2 - 2\sqrt{3}$ is a root, then $x - (2 - 2\sqrt{3})$ is a factor.

Hence: . $P(x) \;=\;(x+1)^2(x - [2+2\sqrt{3}])(x - [2-2\sqrt{3}]) \;=\;0$

And we have: . $(x^2 + 2x + 1)(x^2-4x - 8) \;=\;0$

. . Therefore: . $\boxed{x^4 - 2x^3 - 15x^2 - 20x - 8 \;=\;0}$