Proof of the Remainder Theorem
The remainder theorem states that when a polynomialis divided by a linear expressionthe remainder is
Example: Whenis devider bysubstituteto obtain the remainder
We can prove it quite easily by performing long division ofbyto obtain the quotientand remainderWe can write
If the degree ofisthen the degree ofis and the degree ofmust be one less than the degree ofie the degree ofis 0 sois a constant. We can write
so thatas required.