Polynomial Resultant
Build a Sylvester matrix for two quadratics and test whether they share a complex root.
Free theory articleBuild a Sylvester matrix for two quadratics and test whether they share a complex root.
Concept & interpretation
The resultant of two polynomials is an algebraic invariant built from their coefficients that vanishes if and only if the two polynomials share at least one common root over the complex numbers.
Experiment
You can test the resultant interactively in the Polynomial Resultant module. Try the following steps:
- Set a to 1, b to 1, c to 1, and d to 1 to represent identical polynomials.
- Observe that the Resultant becomes exactly 0.
- The Common complex root indicator updates to true, as identical polynomials share all roots.
- Now change a to 0, b to -1, c to -2, and d to 1.
- The Resultant again evaluates to 0, showing that f(x) = x2 - 1 and g(x) = x2 - 2x + 1 share the real root x = 1.