How did symbolic algebra change mathematical work in the 17th century?

In the 17th century, the development of symbolic algebra fundamentally transformed mathematical work by enabling a powerful synthesis between algebra and geometry, which paved the way for the invention of calculus.12

Key advancements included:

• Systematic Notation: Building on the work of François Viète, who is often called the "father of modern algebraic notation," mathematicians began using letters to represent both known and unknown quantities. This allowed for greater flexibility in symbolic manipulation and the development of a general theory of equations.134
• Analytic Geometry: René Descartes and Pierre de Fermat used these algebraic techniques to solve complex geometric problems that had remained unsolved since antiquity. By representing geometric curves as algebraic equations involving coordinates, they established a new, organic connection between the two fields.12
• Algebraic Properties: Descartes systematically analyzed polynomial equations, exploring the relationship between the degree of an equation and its roots, and introduced methods for manipulating equations that significantly expanded the scope of mathematical reasoning.1

These innovations allowed mathematicians to move beyond the limitations of classical Greek geometry, providing the tools necessary to characterize curves and analyze their properties through algebraic expressions.12