Expressions relating to the Roots of Polynomials#

Rewriting root equations as reciprocals

\[\frac{1}{\alpha}+\frac{1}{\beta}=\frac{\alpha+\beta}{\alpha\beta}$$ $$\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}=\frac{\alpha\beta+\alpha\gamma+\beta\gamma}{\alpha+\beta+\gamma}$$ $$\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}+\frac{1}{\delta}=\frac{\alpha\beta\gamma+\beta\gamma\delta+\alpha\gamma\delta+\alpha\beta\delta}{\alpha\beta\gamma\delta}\]

Sum of the squares equations

Quadratic: $$ \alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta$$
Cubic: $\(\alpha^2+\beta^2+\gamma^2=(\alpha+\beta+\gamma)^2-2(\alpha\beta+\beta\gamma+\alpha\gamma)$\)
Quartic: $\(\alpha^2+\beta^2+\gamma^2+\delta^2=(\alpha+\beta+\gamma+\delta)^2-2(\alpha\beta+\alpha\gamma+\alpha\delta+\beta\gamma+\beta\delta+\gamma\delta)$\)

Sum of the cubes equations

Quadratic: $\(\alpha^3+\beta^3=(\alpha+\beta)^3-3\alpha\beta(\alpha+\beta)$\)
Cubic: $\(\alpha^3+\beta^3+\gamma^3=(\alpha+\beta+\gamma)^3-3(\alpha+\beta+\gamma)(\alpha\beta+\beta\gamma+\alpha\gamma)+3\alpha\beta\gamma$\)

Rules for products of powers

Quadratic: $\(\alpha^n\times\beta^n=(\alpha\beta)^n$\)
Cubic: $\(\alpha^n\times\beta^n\times\gamma^n=(\alpha\beta\gamma)^n$\)
Quartic: $\(\alpha^n\times\beta^n\times\gamma^n\times\delta^n=(\alpha\beta\gamma\delta)^n$\)

Expressions relating to the Roots of Polynomials#

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