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Algebraic notation represents variables, coefficients and algebraic operations by written symbols, so one expression can describe a family of equations.

François Viète worked on algebraic notation and relations between polynomial roots and coefficients.^[ref]

Al-Khwarizmi presented geometric completions of squares in his treatment of quadratic equations. His exposition used verbal instructions, positive lengths and areas, rather than the modern algebraic notation used here. The area assembly above illustrates the same principle with a different expression and arrangement. The modern polynomial identity also covers negative values of the variable and roots of a polynomial in the complex numbers, which need no interpretation as positive side lengths.^[ref]

François Viète studied the connection between positive roots of a polynomial and the coefficients of the powers of the unknown. His 1591 presentation of algebraic notation used letters for both known and unknown quantities, with consonants for known quantities and vowels for unknown quantities. That move makes it possible to express a relation for whole families of polynomials, rather than only a numerical instance. His historical setting concerned positive roots and expressions satisfying dimensional homogeneity; the modern splitting field formulation above is a broader statement.^[ref]