Proof of the Factor Theorem

Letbe a polynomial of degreeis a factor ofif and onlydividesor is a factor of

Proof

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

(1)

The degree ofis less than the degree ofsoso is just a constant. WriteNow substituteinto (1) to give

so thatand we can write