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!
Hello, meli3000!
Determine the polynomial equation in standard form that has the following roots:
. . $\displaystyle -1\:\text{(of order 2)},\;\,2+2\sqrt{3},\;\;2-2\sqrt{3}$
If the polynomial $\displaystyle P(x)$ has root $\displaystyle -1\text{ (order 2)}$, then $\displaystyle (x+1)^2$ is a factor of $\displaystyle P(x)$.
If $\displaystyle 2 + 2\sqrt{3}$ is a root, then $\displaystyle x - (2 + 2\sqrt{3})$ is a factor.
If $\displaystyle 2 - 2\sqrt{3}$ is a root, then $\displaystyle x - (2 - 2\sqrt{3})$ is a factor.
Hence: .$\displaystyle P(x) \;=\;(x+1)^2(x - [2+2\sqrt{3}])(x - [2-2\sqrt{3}]) \;=\;0$
And we have: .$\displaystyle (x^2 + 2x + 1)(x^2-4x - 8) \;=\;0$
. . Therefore: . $\displaystyle \boxed{x^4 - 2x^3 - 15x^2 - 20x - 8 \;=\;0}$