If the question states that the roots of an equation are $\{0,1+i,1+2i\}$ how can I get the equation? As far as I know, most tutorials or website which explain usually have roots where one root is real and two are complex but are conjugates, in which case when we multiply the factors, the imaginary part gets cancelled and you are left with equation in form of $x's$. However in this case we get:
$(x)(x-(1+i)(x-(1+i))) = (x)(x-1-i)(x-1-2i) = (x^3-2x^2-1)+i(3x-3x^2)$
Is there a way to get an equation in a form of polynomial in terms of just x?
Thank you!
Best Answer
Let $r_n$ be the nth root from a set of $N$ roots. Then an equation, that is satisfied by these roots, can be given by: $$\prod_{n=1}^N\left(x - r_n\right) = 0$$ Where the $\prod$ indicates taking the product of all of the terms on the right. This ends up being simply a polynomial in $x$: $$\prod_{n=1}^N\left(x - r_n\right) = (x-r_1)(x-r_2) \cdots (x-r_N) = \sum_{n=0}^Na_n x^n$$ The coefficients of the polynomial, $a_n$, will only be real if the roots are real or come in complex conjugate pairs.
Of course, this isn't the only function that has these roots. There is an infinite set of equations that you could come up with that have these roots and only these roots without additional constraints on the set of allowable functions.