Theory of Equations — Formula Sheet

Theory of Equations — Formula Sheet

Quadratic \(ax^{2}+bx+c=0\)

• Roots: \(x=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}\).
• Discriminant: \(\Delta=b^{2}-4ac\). Real distinct if \(\Delta\gt 0\); equal if \(\Delta=0\); imaginary if \(\Delta\lt 0\).
• Sum of roots \(=-\dfrac{b}{a}\); product of roots \(=\dfrac{c}{a}\).
• Equation from roots: \(x^{2}-(\text{sum})x+(\text{product})=0\).

Vieta’s formulae (monic)

• Cubic \(x^{3}+bx^{2}+cx+d\), roots \(\alpha,\beta,\gamma\):
  \(\alpha+\beta+\gamma=-b\), \(\ \alpha\beta+\beta\gamma+\gamma\alpha=c\), \(\ \alpha\beta\gamma=-d\).
• Quartic \(x^{4}+bx^{3}+cx^{2}+dx+e\), roots \(\alpha,\beta,\gamma,\delta\):
  \(\sum\alpha=-b\), \(\ \sum_{i\lt j}\alpha_i\alpha_j=c\), \(\ \sum_{i\lt j\lt k}\alpha_i\alpha_j\alpha_k=-d\), \(\ \alpha\beta\gamma\delta=e\).
• Degree \(n\), non-monic \(a_{n}x^{n}+\cdots+a_{0}\):
  \(\displaystyle\sum\alpha_i=-\frac{a_{n-1}}{a_{n}},\quad \sum_{i\lt j}\alpha_i\alpha_j=\frac{a_{n-2}}{a_{n}},\quad\ldots,\quad \prod\alpha_i=(-1)^{n}\frac{a_{0}}{a_{n}}.\)

Useful identities on roots

• \(\displaystyle\sum\alpha_i^{2}=\left(\sum\alpha_i\right)^{2}-2\sum_{i\lt j}\alpha_i\alpha_j\).
• \(\displaystyle\sum\frac{1}{\alpha_i}=\frac{\sum_{i\lt j}\alpha_i\alpha_j}{\prod\alpha_i}\) (for a cubic, \(=-\dfrac{c}{d}\)).
• Cube case: \(\alpha^{3}+\beta^{3}+\gamma^{3}-3\alpha\beta\gamma =(\alpha+\beta+\gamma)\!\left(\alpha^{2}+\beta^{2}+\gamma^{2}-\alpha\beta-\beta\gamma-\gamma\alpha\right)\).

Conjugate-root factors

• Imaginary pair (real coefficients): if \(\alpha+i\beta\) is a root, so is \(\alpha-i\beta\), and \(\left(x-(\alpha+i\beta)\right)\left(x-(\alpha-i\beta)\right)=x^{2}-2\alpha x+\left(\alpha^{2}+\beta^{2}\right)\).
• Surd pair (rational coefficients): if \(p+\sqrt{q}\) is a root, so is \(p-\sqrt{q}\), and \(\left(x-(p+\sqrt q)\right)\left(x-(p-\sqrt q)\right)=x^{2}-2px+\left(p^{2}-q\right)\).

Rational Root Theorem

• For integer coefficients, a rational root \(\dfrac{p}{q}\) (lowest terms) satisfies \(p\mid a_{0}\) and \(q\mid a_{n}\).
• Monic case: every rational root is an integer dividing \(a_{0}\).

Reciprocal equations

• Type I: \(a_{k}=a_{n-k}\). Type II: \(a_{k}=-a_{n-k}\).
• Reduction (even degree): set \(y=x+\dfrac{1}{x}\) (Type I) or \(y=x-\dfrac{1}{x}\) (Type II), noting \(x^{2}+\dfrac{1}{x^{2}}=y^{2}-2\) and \(x^{2}+\dfrac{1}{x^{2}}=y^{2}+2\) respectively.

Descartes’ Rule of Signs

• Positive roots \(=s,\,s-2,\,s-4,\ldots\), where \(s\) = sign changes in \(P(x)\).
• Negative roots: apply the same to \(P(-x)\).
• Imaginary roots (lower bound): \(n-(\text{max positive})-(\text{max negative})\), adjusted for any forced \(0\) root.