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).

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.