# Understanding Polynomial Equations in *n* Unknowns

# Understanding Polynomial Equations in *n* Unknowns

This chapter examines the search for simultaneous solutions which eventually led to the creation of whole new algebraic constructs over the course of the eighteenth and nineteenth centuries. The constructs in question were the determinants, matrices, and invariants, among others. These, like groups, came to define new algebraic areas of inquiry in and of themselves. They were borne through a protracted search over the latter half of the millennium for more general techniques for solving systems of equations simultaneously—an issue which has been coming up frequently in mathematical problems since ancient times. The chapter looks at the evolution of these constructs and how mathematicians had come ever closer to their ideal solution.

*Keywords:*
polynomial equations, n unknowns, simultaneous solutions, new algebraic constructs, determinants, matrices, invariants, linear equations, linear transformations

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