Question

Consider the following two polynomials: $Poly1: \lambda^{n}+a_{1} \lambda^{n-1}+a_{2} \lambda^{n-2}+ a_{3} \lambda^{n-3} + \cdots \quad poly2: \lambda^{n}-a_{1} \lambda^{n-1}+a_{2} \lambda^{n-2}-a_{3} \lambda^{n-3}+\cdots$

i.e., the magnitudes of the coefficients of each power of $\lambda$ are identical in both the polynomials but the signs alternate. For such polynomials, is there any relationship between the roots of these polynomials?

Thank you.

Answer 1

If I am not missing something in your question, this looks simple: $\lambda_{poly1} = - \lambda_{poly2}$...

Answer 2

Well, if n is even, then it seems like $$p_1(x) = p_2(-x)$$ and hence $x$ is a root of $p_1$ iff $-x$ is a root of $p_2.$

Answer 3

if $p_1(x)=\sum_k b_k x^k=0$, then $$\pm p_2(-x)=\sum_k (-1)^k b_k (-x)^k=\sum_k b_k x^k=0$$

Therefore this is indeed most probably homework $\smile$...