Synthetic division is a numerical method for dividing a polynomial by a binomial of the form x – c. The technique involves writing only the essential parts of a long division problem.

The following illustrates the use of synthetic division to divide a polynomial, written in descending order, by x – c , where c is a constant called the divider:

To divide a polynomial f(x) by x – c:	Divide: 2x - 7x + 7x – 13 by x - 3
1. Write the divider, 3, and the coefficients of the dividend
2. Bring the first coefficient of the dividend down to the bottom row.
3. Multiply the first number in the bottom row by the divider; write this result in the 2nd column of the middle row
4. Add the 2nd column to obtain the 2nd number in the bottom row
5. Similarly, multiply the –1 in the bottom row by the divider; write this result in the 3rd column of the middle row; then add the 3rd column to obtain 4.
6. Finally, multiply 4 by the divider; write this result in the 4th column of the middle row; then add the 4th column and write the result in the bottom row.
7. The entries in row three, excluding the last entry, are the coefficients of the quotient. The degree of the quotient is one less than the degree of the dividend. The last entry is the remainder of the division.	The degree of the dividend is 3. So the degree of the quotient is 2. Quotient: 2x- x + 4 Remainder: -1 2x- x + 4 -

Example. Use synthetic division to perform the indicated operation:

Solution. The divisor, x + 2, must be written as a difference, x – (-2), to determine that the divider is –2.

The Remainder Theorem

If a polynomial f(x) is divided by x – c , where c is a constant, then the remainder is equal to f(c).

Note: The Remainder Theorem applies only if the divisor is of the form x – c. Divisors of the form x + c must be changed to x – (-c).

The Remainder Theorem gives us a second way to evaluate a polynomial for x = c. We can evaluate f(c) directly, or we can use synthetic division.

Example. Use the Remainder Theorem and synthetic division to find f(k):

k = 4 ; f(x) =

Solution. The Remainder Theorem tells us that if we use synthetic division and divide f(x) with divider 4, the remainder will be equal to f(4).

The dividend has no "x" term; the coefficient 0 must be used for this missing term.

The remainder is 127. So, f(4) = 127.

Example. Use synthetic division to decide whether -6 is a zero of the polynomial

f(x) = .

Solution. We know that if -6 is a zero of f(x), then f(-6) = 0. So, we will use synthetic division and the Remainder Theorem to find f(-6).

Since the remainder is 0, f(-6) = 0, and –6 is a zero of the polynomial f(x).

Division Algorithm

If f(x) and d(x) are polynomials, with d(x) ¹ 0, and the degree of d(x) is less than or equal to the degree of f(x), then there exist unique polynomials q(x) and r(x) such that

f(x) =	d(x) ×	q(x) +	r(x)
Dividend =	Divisor ×	Quotient +	Remainder

If r(x) = 0, then d(x) divides evenly into f(x), and d(x) and q(x) are factors of f(x).

We will use the Division Algorithm to do the following problem:

Express the polynomial f(x) = in the form f(x) = (x – k )q(x) + r for k = 2.

In this case, the divisor, d(x), is x – 2. We will use synthetic division to divide f(x) by

x – 2 .

The quotient, q(x), is , and the remainder, r, is 6.

f(x) = (x – 2 )( ) + 6