Aus Online Mathematik Brückenkurs 2

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

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

\displaystyle \left( z-1 \right)\left( z-2 \right)\left( z-4 \right)=z^{3}-7z^{2}+14z-8

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