Roots of a Cubic Equation
The big rules! (Cubic)
- \(\alpha+\beta+\gamma=-\frac{b}{a}\)
- \(\alpha\beta+\alpha\gamma+\beta\gamma=\frac{c}{a}\)
- \(\alpha\beta\gamma=-\frac{d}{a}\)
The equation for a cubic
\(\(ax^3+bx^2+cx+d=\)\)(\(a(x-\alpha)(x-\beta)(x-\gamma)\)\)
We expand this out to get:
- \(\(a(x-\alpha)(x^2-x\gamma-x\beta+\beta\gamma)\)\)
- (\(a(x^3-x^2\gamma-x^2\beta+x\beta\gamma-x^2\alpha+x\alpha\gamma+x\gamma\beta-\alpha\beta\gamma)\)\)
\(x^3+\frac{b}{a}x^2+\frac{c}{a}x+\frac{d}{a}\)
- \(\(x^3-x^2\gamma-x^2\alpha-x^2\beta+x\beta\gamma+x\alpha\gamma+x\alpha\beta-\alpha\beta\gamma\)\)
We then factorise out: -
\[x^3-(\alpha+\beta+\gamma)x^2+(\alpha\beta+\alpha\gamma+\beta\gamma)x-\alpha\beta\gamma\]