Proof of the Factor Theorem
Letbe a polynomial of degreeis a factor ofif and onlydividesor is a factor of
Letbe a polynomial and letbe a number.
Ifdividesthen the remainder on division ofbyis zero and there is a polynomialsuch thatso thatandis a root of
Now assume thatis a root ofso thatPerform long division ofbyto obtain quotientand remainderthen write
The degree ofis less than the degree ofsoso is just a constant. WriteNow substituteinto (1) to give
so thatand we can write