Depressed Quartic Polynomials -Translation Math Simplified

An ‘Intuitive’ Graphical Approach to Depress Quartics Without Lengthy Multiplication

Cardano’s famous book Ars Magna published in 1545 presented various solutions for finding Cubic and Quartic polynomial roots involving depressing the functions to simplify the math. For an informative read on the history of the publication I suggest Kasper Muller’s The Battle of the Cubic Equation and the Dawn of the Complex Numbers.

Depressing polynomials requires shifting the function horizontally by (x+-u) units to derive a simpler function by elimination of the 2nd term. For example, Cubic polynomial: y=Ax³+Bx²+Cx+D reduces to a new Cubic y=ax³+cx+d (which has no bx² content).

In the case of a Quadratic, u=-b/2a represents the turning point, whereas with the higher orders, u gives the Inflection Point/s.

To depress a Quartic such as y=Ax⁴+Bx³+Cx²+Dx+E requires lengthy multiplication of y=A(x+-u)⁴+B(x+-u)³+C(x+-u)²+D(x+-u)+E to derive y=ax⁴+cx²+dx+e.

This post avoids the lengthy multiplication process. Instead, it simply relies on the fact that polynomial graphs retain their shapes under horizontal translation. So by knowing original intercept coordinates in the x-y grid, it is a simple matter to…