Aus Online Mathematik Brückenkurs 2

There exists a simple relation between a zero and the polynomial's factorization: \displaystyle z=a is a zero if and only if the polynomial contains the factor \displaystyle (z-a). (This is the meaning of the factor theorem.)

If we are to have a polynomial with zeros at \displaystyle 1, \displaystyle 2 and \displaystyle 4, the polynomial must therefore contain the factors \displaystyle (z-1), \displaystyle (z-2) and \displaystyle (z-4). For example,

\displaystyle (z-1)(z-2)(z-4) = z^3-7z^2+14z-8\,\textrm{.}

Note: It is possible to multiply the polynomial above by a non-zero constant and get another third-degree polynomial with the same roots.