Let us continue our exploration of polynomials. Just as in the previous blog, let me present to you an outline of some method(s) to solve Quartics. As I said earlier, “filling up the gaps” will kindle your intellect. Above all, the main aim of all teaching is teaching “to think on one’s own”.

**I) The Quartic Equation. Descartes’s Method (1637).**

**(**a) Argue that any quartic equation can be solved once one has a method to handle quartic equations of the form:

(b) Show that the quartic polynomial in (a) can be written as the product of two factors

where u, v, w satisfy the simultaneous system

Eliminate v and w to obtain a cubic equation in .

(c) Show how any solution u obtained in (b) can be used to find all the roots of the quartic equation.

(d) Use Descartes’s Method to solve the following:

**II) The Quartic Equation. Ferrari’s Method:**

(a) Let a quartic equation be presented in the form:

The strategy is to complete the square on the left side in such a way as to incorporate the cubic term. Show that the equation can be rewritten in the form

where u is indeterminate.

(b) Show that the right side of the transformed equation in (a) is the square of a linear polynomial if u satisfies a certain cubic equation. Explain how such a value of u can be used to completely solve the quartic.

(c) Use Ferrari’s Method to solve the following:

.

Happy problem-solving 🙂

Do leave your comments, questions, suggestions…!

More later…

Nalin