Sylvester's matrix

Sylvester's matrix is a special construction used in algebra to determine if two polynomial expressions share a common root. It’s built from the coefficients of the polynomials and arranged into a larger matrix. When you calculate the determinant (a single number) of this matrix, if it equals zero, the polynomials have at least one root in common; if not, they do not. This method provides a systematic and efficient way to analyze relationships between polynomial equations without explicitly solving them.