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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.

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You can use Newton's identities.

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 $$p_2=c_{1}^2-2c_{2}$$

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