ax2+bx+c=0

Abstract This chapter is about solving polynomial equations. A ‘polynomial’ is an algebraic expression like 3x5+4x4-8x2−5x+6, obtained by adding and subtracting numbers and powers of a variable quantity (x), and a ‘polynomial equation’ is an equation like 3x5+4x4-8x2-5x+6=0; its ‘degree’ is the exponent of the highest power (here, 5). The aim is to find values of x that satisfy them, such as x=1 in this example. After describing a range of examples, from Ancient Egypt, Mesopotamia, Greece, and China, of linear equations (equations of degree 1, of the form ax=b), this chapter turns to quadratic equations (equations of degree 2, of the form ax2+bx+c=0) and their solution by ‘completing the square’, as developed by Islamic mathematicians in the 9th century ad. Later, in the 16th century, methods were presented for solving cubic equations (equations of degree 3) by the Italians Gerolamo Cardano and Niccolò Tartaglia, and their methods were then extended to the solution of equations of degree 4. The question then arose as to whether similar methods could be found for solving polynomial equations of degree 5 or more, and the chapter concludes with a discussion of how this was answered.