Vieta's Pattern for Higher-Degree Polynomials
Vieta's Pattern for Higher-Degree Polynomials
By the Fundamental Theorem of Algebra, a degree-\(n\) polynomial equation has exactly \(n\) roots, counted with repetition and including complex roots. The quadratic and cubic results extend to every degree, with one alternating pattern of signs.
| Equation | \(\sum\alpha\) | \(\sum\alpha\beta\) | \(\sum\alpha\beta\gamma\) | \(\sum\alpha\beta\gamma\delta\) |
|---|---|---|---|---|
| \(ax^2+bx+c=0\) | \(-\dfrac{b}{a}\) | \(\dfrac{c}{a}\) | ||
| \(ax^3+bx^2+cx+d=0\) | \(-\dfrac{b}{a}\) | \(\dfrac{c}{a}\) | \(-\dfrac{d}{a}\) | |
| \(ax^4+bx^3+cx^2+dx+e=0\) | \(-\dfrac{b}{a}\) | \(\dfrac{c}{a}\) | \(-\dfrac{d}{a}\) | \(\dfrac{e}{a}\) |
In general, for \(a_nx^n+a_{n-1}x^{n-1}+\dots+a_1x+a_0=0\) with roots \(r_1,\ r_2,\ \dots,\ r_n\): the sum of roots is \(-\dfrac{a_{n-1}}{a_n}\) and the product of all \(n\) roots is \((-1)^n\dfrac{a_0}{a_n}\).
The extra negative signs appear alternately, starting from \(-\dfrac{b}{a}\); whether the product of all the roots carries a negative sign depends on whether \(n\) is odd or even.
\(\sum\alpha\beta\gamma\) means the sum of ALL products of three roots chosen from distinct positions in the root list (values may repeat). With four roots \(\alpha,\ \beta,\ \gamma,\ \delta\) it has four terms: \(\alpha\beta\gamma+\alpha\beta\delta+\alpha\gamma\delta+\beta\gamma\delta\).
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