A Polynomial Function with an Irreducible Factor
This Demonstration illustrates how a polynomial function, with real zeros of any multiplicity, changes when multiplied by a polynomial irreducible over the reals, , where and are two coefficients that serve to modify the shape of the function. When , we simply add another real zero of the function and there are no complex solutions, but when we begin increasing , a maximum with a couple of minima (or a minimum with a couple of maxima, depending on the function) starts moving until the three points merge to a single point, giving a sketching graph like as the function without any irreducible polynomial. Flex point is initially horizontal, then oblique and finally disappears further increasing b value. This happens when exceeds a critical value with the general condition , if . A vertical shifting of the maximum (or minimum) occurs when is equal to the value of the maximum (or minimum) of the function without the irreducible polynomial; otherwise, the shifting will follow an oblique trajectory (indicated with a red line).[more]
Two functions are considered: and . In the first case, the point set is (indicated by a black line), which becomes a relative minimum. In the second case, the point becomes a relative maximum. Use the controls to change the shape of the function by modifying and values.[less]
Snapshot 1: the presence of an irreducible polynomial adds two relative minima and a relative maximum before the value reaches the critical value
Snapshot 2: the presence of an irreducible polynomial adds two relative maxima and a relative minimum before the value reaches the critical value
Snapshot 3: after the critical value of , the shape of the function follows the same trend as the original function: one relative minimum without any flex
 E. Perano, Studio immediato di funzioni, Italy: Clut,2020.