The Remainder and Factor Theorems - Chapter 6 Review: Section 5

Polynomial long division can be used to divide one polynomial by another when the dividend has a higher degree than the divisor.
Under the division sign, write the dividend polynomial in standard form. Outside it, write the divisor. Make sure to include "absent" terms by giving them a zero coefficient.
What times the first term of the divisor would equal the first term of the dividend? Write it above the term of the dividend with the same exponent.
Write the product of the divisor and the number just written over the division bar underneath the dividend.
Subtract like in regular long division.
Carry down the next part of the dividend like in regular long division.
Treat the polynomial furthest down the page like it were the dividend and start again at step 3.
When the degree of the divisor becomes larger than the degree of the "pseudo-dividend," the pseudo-dividend is the remainder. Write the remainder to the right of the quotient and put it over the divisor.

Because of something called the Remainder Theorem, when the divisor is in the form (x - k), then you can use synthetic division to divide polynomials.
Synthetic division works just like synthetic division as discussed in Section 2.
Take k (remember it is negative in (x - k) so you need to switch the sign) and use it as the x value you are evaluating for in your function.
The numbers under the line are the coefficients of the quotient polynomial, in descending order of degrees. The degree of the quotient will always be one less than the degree of the dividend, since you can only divide out linear x's (x - k) with synthetic division.
The number in the box at the end is the remainder. Remember to put it over the divisor when writing your answer.

The Factor Theorem says that if the remainder of a polynomial quotient is zero, then the divisor was a factor of the polynomial. Also, the function for the value of k of the factor will be a zero.
Because of this, you can factor a polynomial using synthetic division.

Divide.

Find all the zeroes.