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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 ) = x² – 8 x + 6. Divide the polynomial by the binomial x – 2.

We can do the division in either method.

Method 1: Long Division

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The remainder is –6.

Method 2: Synthetic Division

The remainder is –6.

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

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 , where q ( x ) is a polynomial with degree one less than the degree of f ( x ).

In other words, dividend equals quotient times divisor plus remainder.

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.