Polynomials

VMM

May 2010
3
0
What is the degree of the simplest polynomial with the zeroes:√2, 2√2 + i, and 2 + 2i ??
 
May 2010
43
3
Norman, OK
My guess would be 6 since the simplest polynomial that I can come up with that has all of these as roots is (x² - 2)(x² - 4√2 x + 9)(x² - 4x + 8).
 
Nov 2009
927
260
Wellington
This means that \(\displaystyle x - \sqrt{2}\), \(\displaystyle x - (2 \sqrt{2} + \pi)\), \(\displaystyle x - (2 + 2 \pi)\) are all factors of this polynomial. Therefore the simplest polynomial that has all these roots is :

\(\displaystyle (x - \sqrt{2})(x - (2 \sqrt{2} + \pi))(x - (2 + 2 \pi)) = 0\)

Or, equivalently :

\(\displaystyle (x - \sqrt{2})(x - 2 \sqrt{2} - \pi)(x - 2 - 2 \pi) = 0\)

And the degree of this polynomial is, trivially, 3 (can be checked by expanding).