When attempting to find the roots of a polynomial, it will be useful to be able to divide that polynomial by other polynomials. Here we'll learn how.

Long division of polynomials is a lot like long division of real numbers. If
the polynomials involved were written in fraction form, the numerator would be
the dividend, and the denominator would be the divisor. To divide polynomials
using long division, first divide the first term of the dividend by the first
term of the divisor. This is the first term of the quotient. Multiply the new
term by the divisor, and subtract this product from the dividend. This
difference is the new dividend. Repeat these steps, using the difference as the
new dividend until the first term of the divisor is of a greater degree than
the new dividend. The last "new dividend" whose degree is less than that of the
divisor is the remainder. If the remainder is zero, the divisor divided evenly
into the dividend. In the example below,
*f* (*x*) = *x*
^{4} +4*x*
^{3} + *x* - 10
is
divided by
*g*(*x*) = *x*
^{2} + 3*x* - 5
.

Figure %: Long division of polynomials.

Two important theorems pertain to long division of polynomials.

The Remainder Theorem states the following: if a polynomial
*f* (*x*)
is divided
by the polynomial
*g*(*x*) = *x* - *c*
, then the remainder is the value of
*f*
at
*c*
,
*f* (*c*)
.

The Factor Theorem states the following: Let
*f* (*x*)
be a polynomial;
(*x* - *c*)
a factor of
*f*
if and only if
*f* (*c*) = 0
. This means that if a given value
*c*
is a root of a polynomial, then
(*x* - *c*)
is a factor of that polynomial.

Synthetic division is an easy way to divide polynomials by a polynomial of the
form
(*x* - *c*)
. It is both a way to calculate the value of a
function at
*c*
(Remainder Theorem) as well as
to check whether or not
*c*
is a root of the polynomial (Factor Theorem).
Synthetic division is a shortcut to long division. It requires only three lines
-- the top line for the dividend and divisor, the second line for the
intermediate values, and the third line for the quotient and remainder. It is
done this way. Let the dividend have degree
*n*
. 1) In line one write the
coefficients of the polynomial as the dividend, and let
*c*
be the divisor. 2)
In line three rewrite the leading coefficient of the dividend directly below
its position in the dividend. 2) Multiply it by the divisor, and write the
product in line two directly below the coefficient of
*x*
^{n} - 1
. 3) Add this
product to the number directly above it in the dividend (this number is the
coefficient of
*x*
^{n} - 1
) to get a new number. Repeat steps two and three until
the entire polynomial has been divided. The quotient will be one degree less
than the dividend. The coefficients of the quotient are the first
*n* - 1
numbers in line three. The remainder is the last number in line three. Below a
polynomial of the form
(*x* - *c*)
is divided using long division, and then using
synthetic division. Study it carefully.

Figure %: Long division of a polynomial of the form
(*x* - *c*)
.

Figure %: Synthetic division.