# Powers of roots in terms of polynomial coefficients

Suppose we have a monic polynomial of degree $$n$$ with coefficients $$c_1, c_2, c_3, \cdots, c_n$$, and roots $$r_1, r_2, r_3, \cdots, r_n$$: $$x^n+c_1 x^{n-1} + c_2 x^{n-2} + c_3x^{n-3} + \cdots + c_n$$

I'm looking to find expressions such as $$r_1^2 + r_2^2 + r_3^2 + \cdots + r_n^2 \\ r_1^3 + r_2^3 + r_3^3 + \cdots + r_n^3 \\ r_1^4 + r_2^4 + r_3^4 + \cdots + r_n^4 \\$$ in terms of the coefficients $$c_k$$.

I already know how to do the first few on a case by case basis, so I'm looking for a more general solution or method for handling higher powers and higher degree polynomials, if they exist.

I suspect there's some simple inductive method I'm just not seeing.

• – Gary Mar 18 at 18:08

This process would be inductive. The coefficient of $$x^{n-k}$$ is $$(-1)^ke_k$$ by the notation in the article on Newton's identities. Your desired sums are $$p_k=r_1^k+r_2^k+\cdots+r_n^k$$
Then the formula says $$ke_k=e_{k-1}p_1-e_{k-2}p_2-e_{k-3}p_3+\cdots+(-1)^{k-1}p_k$$ Substituting in the coefficients, you can solve for $$p_k$$. For example, $$p_1=e_1=-c_{1}$$ $$2e_2=2c_{2}=e_1p_1-p_2=c_{1}^2-p_2$$ so $$p_2=c_{1}^2-2c_{2}$$