The Remainder Theorem
Generally when a polynomial is divided by a binomial there is a remainder.
Consider the polynomial function . Divide the polynomial by the binomial .
We can do the division in either method.
Method 1: Long Division
The remainder is .
Method 2: Synthetic Division
The remainder is .
Now compare the remainder of to .
Notice that the value of is the same as the remainder when the polynomial is divided by the binomial . This illustrates the Remainder Theorem.
If a polynomial is divided by , the remainder is the constant , and , where is a polynomial with degree one less than the degree of .
Synthetic division is a simpler process for dividing a polynomial by a binomial. When synthetic division is used to evaluate a function, it is called synthetic substitution.