FP1 Topic 3: Complex numbers
Roots of cubics
The cubic equation ax^3+bx^2+cx+d=0 has roots \alpha, \beta and \gamma, therefore \displaystyle x^3+\frac{b}{a}x^2 + \frac{c}{a}x + \frac{d}{a} \equiv (x-\alpha)(x-\beta)(x-\gamma) therefore \displaystyle x^3+\frac{b}{a}x^2 + \frac{c}{a}x + \frac{d}{a} \equiv x^3-(\alpha+\beta+\gamma)x^2+(\alpha\beta+\alpha\gamma+\beta\gamma)x-\alpha\beta\gamma therefore \alpha+\beta+\gamma = -\displaystyle \frac{b}{a} , \alpha\beta+\alpha\gamma+\beta\gamma = \displaystyle\frac{c}{a} and \alpha\beta\gamma = -\displaystyle\frac{d}{a}

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