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Long division is useful with the remainder and factor theorems, but long
division can be time consuming. To divide a polynomial by a binomial and
compute the remainder, we can also use synthetic division. We can only
divide by a binomial whose leading
coefficient is 1--thus, we must factor the
leading coefficient out of the binomial and divide by the leading coefficient
separately. Also, the binomial must have degree 1; we cannot use synthetic
division to divide by a binomial like x^{2} + 1. Here are the steps for
dividing a polynomial by a binomial using synthetic division:

Write the polynomial in descending
order, adding "zero terms" if an exponent
term is skipped.

If the polynomial does not have a leading coefficient of 1, write the
binomial as b(x - a) and divide the polynomial by b. Otherwise, leave the
binomial as x - a.

Write the value of a, and write all the coefficients of the polynomial in
a horizontal line to the left of a.

Draw a line below the coefficients, leaving room above the line.

Bring the first coefficient below the line.

Multiply the number below the line by a and write the result above the
line below the next coefficient.

Subtract the result from the coefficient above it.

Repeat steps 6 and 7 until all the coefficients have been used.

If the polynomial has n terms, the first n - 1 numbers below the line
are the coefficients of the resulting polynomial, and the last number is the
remainder.

Example: What is the result when 4x^{4} -6x^{3} -12x^{2} - 10x + 2 is
divided by x - 3? What is the remainder?

The result is 4x^{3} +6x^{2} + 6x + 8, and the remainder is 26.