Useful identities with sum and product

Many symmetric expressions in $x$ and $y$ can be rewritten in terms of the sum $S=x+y$ and the product $P=xy$. Having them to hand speeds up the solving of symmetric systems.

Observation — Useful identities

$x^2+y^2 = S^2-2P, \qquad x^3+y^3 = S^3-3SP.$

Thanks to these identities, even equations that at first sight seem to contain powers of the unknowns turn into equations in $S$ and $P$ alone.

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Topics: Second-degree equations
Concepts: Algebraic identities · Symmetric system
Skills: Using formulas