ABSTRACT
We are all familiar with the way in which the quadratic formula gives us an explicit formula for the roots of any degree two polynomial in Q[x] (and, this formula works in C[x] too – see Exercises 9.1 and 9.2). In Exercises 9.12-9.19, you can explore the cubic formula that performs the same task as the quadratic formula; it is a good bit more complicated. In the case of the quadratic formula, we need to perform the field operations from Q, and in addition extract a square root. In the case of the cubic formula, we need to perform the field operations from Q, extract cube roots, and also have to extract one or more square roots. Can this process be extended to higher and higher degree polynomials from Q[x]? The answer for quartic (that is, fourth degree) equations is yes; it is a nightmare of a formula, which involves not only the extraction of a fourth root, but also cube and square roots as well. In essence, the cubic problem reduces finally to a quadratic, and the quartic in turn finally reduces to a cubic! You can explore the quartic formula in Exercise 9.20.
