Depressed Quartic Polynomials -Translation Math Simplified

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

Depressing polynomials shifts 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).

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. It simply relies on the fact that, polynomial functions retain their shapes under horizontal translation. So by knowing original intercept coordinates in the X-Y Grid, it is a simple matter to adjust these by u to derive the formula of the new depressed equation.

This method draws on knowledge of Quartic architecture, which is more helpful than calculator-like multiplication when designing your own functions.

I recently posted, The Graph Behind The Math of Quadratic Equations, to graphically demonstrate the process of finding roots by ‘Completing the Square’ which produces a Depressed Quadratic version y=ax²+c.