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Problem

For a cubic polynomial $a_3 x^3 + a_2 x^2 + a_1 x + a_0$, the roots $r_1$, $r_2$, and $r_3$ come in three symmetric combinations: $\sigma_1 = r_1 + r_2 + r_3$, $\sigma_2 = r_1 r_2 + r_1 r_3 + r_2 r_3$, and $\sigma_3 = r_1 r_2 r_3$.

The sum of the $k^{\rm th}$ powers of the roots is defined as $s_k = r_1^k + r_2^k + r_3^k$.

First express $s_3$ in terms of $\sigma_1$, $\sigma_2$, and $\sigma_3$. Then use that answer to factor $x^3 + a^3 + b^3 - 3abx$. (By the way, this is an important step in one derivation of the cubic formula. First transform to $x^3 + px + q$, then let $p = -3ab$ and $q = a^3 + b^3$.)

Details
Contributer: Josh
Authors
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References
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VARIABLES
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DEFINITIONS
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