Given some roots : $r_1,r_2,\ldots,r_n$, how can we reconstruct polynomial coefficients?
I know the Horner scheme and that we can just go backwards receiving those coefficients.
But I'm curious if there's some other nice solution to this problem.
Given some roots : $r_1,r_2,\ldots,r_n$, how can we reconstruct polynomial coefficients?
I know the Horner scheme and that we can just go backwards receiving those coefficients.
But I'm curious if there's some other nice solution to this problem.
Knowing the location of roots really only pins down a family of polynomials, which vary by multiplicative constants. If you know the roots of $P(x)$ are $r_1, r_2, \dots, r_n $ then you can write the polynomial as $ P(x) = a (x-r_1)(x-r_2) \cdots (x-r_n) $
where $a$ is some non-zero constant. However, we can not be more precise than that. We can pin down $a$ if we know the value of $P$ are any other point. If we knew $a$, then simply expanding the right hand side would produce the polynomial in a form where you could read off the constants.
The polynomial is $P(z) = a_0 (z-r_1)\times (z-r_2) \times \cdots \times (z-r_n)$. Now labeling coefficients as $P(z) = a_0 z^n + a_1 z^{n-1} + \ldots + a_n$ we have $ a_1 = -a_0 ( r_1 + r_2 + \ldots + r_n) \qquad a_n = (-1)^n a_0 r_1 \times r_2 \times \cdots \times r_n $ In general: $ a_k = \frac{(-1)^k}{k!} a_0 \sum_\pi r_{\pi(1)} \times \cdots \times r_{\pi(k)} $ where the sum is over permutations $ \pi \in \mathfrak{S}_n$.