For a polynomial with real coefficients, complex roots come in conjugate pairs, that is: if and only if .
Roots and Coefficients
Let , , , and be the roots of the following polynomials (as necessary according to the degree of each). The following relationships are known as Vieta's formulae.
Quadratics
For a quadratic :
Cubics
For a quartic :
Quartics
For a quartic :
Using Substitutions
If an equation in has a root , a substitution can be applied, giving a resulting equation in having a root .
Example
The equation has roots and . Find a quadratic equation with roots and .