$$ \text{Chapter 1: Roots of polynomial equation} $$

• Summation notation of roots $[\alpha, \beta,\gamma,\delta]$$=[$alpha, beta, gamma, delta$]$

<aside> ❄️ $\sum\alpha=\underbrace{\overbrace{\alpha+\beta}^{\text{quadratic}}+\gamma}_{\text{cubic}}+\delta$; Sum of all roots

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<aside> ❄️ $\sum\alpha\beta=\underbrace{\overbrace{\alpha\beta}^{\text{quadratic}}+\alpha\gamma+\beta\gamma}_{\text{cubic}}+\alpha\delta+\beta\delta+\gamma\delta$; Sum of products of pairs of roots

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<aside> ❄️ $\sum\alpha\beta\gamma=\underbrace{\alpha\beta\gamma}_{\text{cubic}}+\alpha\beta\delta+\alpha\gamma\delta+\beta\gamma\delta$; Sum of products of trios of roots

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<aside> ❄️ $\displaystyle S_n=\underbrace{\overbrace{\alpha^n+\beta^n}^{\text{quadratic}}+\gamma^n}_{\text{cubic}}+\delta^n$; Recurrence form

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<aside> ❄️ $\displaystyle S_{-1}=\sum \frac{1}\alpha$

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• Equation has complex roots if $S_2<0$

Quadratics: $\displaystyle ax^2+bx+c=0$

<aside> ❄️ $\displaystyle\sum\alpha=-\frac{b}{a}$

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<aside> ❄️ $\displaystyle \sum\alpha\beta=\frac{c}{a}$

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<aside> ❄️ $\displaystyle S_2=\sum\alpha^2=(\alpha+\beta)^2-2\alpha\beta=\big(\sum\alpha\big)^2-2\sum\alpha\beta$; to find $S_2$

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Cubics: $\displaystyle ax^3+bx^2+cx+d=0$

<aside> ❄️ $\displaystyle \sum\alpha=-\frac{b}{a}$

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<aside> ❄️ $\displaystyle\sum\alpha\beta=\frac{c}{a}$

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<aside> ❄️ $\displaystyle\sum\alpha\beta\gamma=-\frac{d}{a}$

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<aside> ❄️ $\displaystyle S_2=\sum\alpha^2=(\alpha+\beta+\gamma)^2-2\sum\alpha\beta=\big(\sum\alpha\big)^2-2\sum\alpha\beta$

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• $\sum \alpha^3=(\sum \alpha)^3-3\sum\alpha\beta\sum\alpha+3\sum\alpha\beta\gamma$; not recommended to use

Quartic: $\displaystyle ax^4+bx^3+cx^2+dx+e=0$

<aside> ❄️ $\displaystyle \sum\alpha=-\frac{b}{a}$

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<aside> ❄️ $\displaystyle\sum\alpha\beta=\frac{c}{a}$

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<aside> ❄️ $\displaystyle\sum\alpha\beta\gamma=-\frac{d}{a}$

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<aside> ❄️ $\displaystyle\sum\alpha\beta\gamma\delta=\frac{e}{a}$

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