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The Remainder Theorem

Generally when a polynomial is divided by a binomial there is a remainder.

Consider the polynomial function f ( x ) = x 2 8 x + 6 . Divide the polynomial by the binomial x 2 .

We can do the division in either method.

Method 1: Long Division

x 2 x 6 x 2 8 x + 6 x 2 2 x _ 6 x + 6 6 x + 12 _ 6

The remainder is 6 .

Method 2: Synthetic Division

The remainder is 6 .

Now compare the remainder of 6 to f ( 2 ) .

f ( 2 ) = ( 2 ) 8 ( 2 ) + 6 = 4 16 + 6 = 6

Notice that the value of f ( 2 ) is the same as the remainder when the polynomial is divided by the binomial x 2 . This illustrates the Remainder Theorem.

If a polynomial f ( x ) is divided by x a , the remainder is the constant f ( a ) , and f ( x ) = q ( x ) ( x a ) + f ( a ) , where q ( x ) is a polynomial with degree one less than the degree of f ( x ) .

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.