Question

polynomial function of the lowest possible degree witht these, 1+2i,-1, and 2

Answer 1

Since 1 + 2i is a zero, 1 - 2i must also be a zero
You function will be a 4th degree polynomial and will look like this:
f(x) = (x - (1+2i))(x - (1-2i))(x + 1)(x - 2)

Unfortunately you'll have to expand all of that out probably and that's kinda a pain.

Answer 2

First thing is to figure out what the lowest possible degree is. Remember that complex roots such as \(1+2i\) come in complex conjugate pairs \((a\pm bi)\) if you have a polynomial with only real coefficients

Answer 3

It's a matter of just distributing everything out and combining like terms.
If allowed you could also just go to wolframalpha.com and input
expand (x - (1+2i))(x - (1-2i))(x + 1)(x - 2)

Answer 4

thanks

Answer 5

I didn't know all complex roots have to come in pairs!?
Does that mean a + i = a (+-) i ?

Answer 6

complex roots only have to come in pairs if you don't have complex numbers as coefficients in your polynomial. it's perfectly possible to have a polynomial with only one complex number as a root, but it may look a bit odd, something like \[(x-i)(x-1) = x^2 -i x - x +i\]
That is a polynomial with roots \(x=i\) and \(x =1\)

When you have them in the conjugate pairs \(a\pm bi\) the complex part goes away because you have a difference of squares:
\[(a+bi)(a-bi) = a^2 -abi + abi -b^2i^2 = a^2-b^2i^2\]but \(i^2=-1\) so\[a^2-b^2i^2 = a^2-b^2(-1) = a^2+b^2\]look ma, no complex numbers! :-)

Answer 7

Yeah, we used complex conjugate multiplication a lot in AC circuits to simplify fractions in the form a + bi / c + di to get rid of denominator complex numbers. Thanks a lot for your description! Plus points!

Answer 8

you probably made the bottom of your i's a bit curvy, almost so they looked like j's, right? :-)

Answer 9

Correct, so we could not confuse current symbol represented by the letter i they chose to use j and put it in front of the number as such. 2 + j2

Answer 10

Yep, I was a EE long ago :-)

Answer 11

I'm sadly thinking of switching to CS.

Answer 12

I don't think the math is any easier, but less chance of injuring yourself with molten metal, perhaps :-)