Long division of a polynomial by a binomial is carried out in essentially the same manner as long division of two integers with no variables:

1. Divide the highest degree term of the polynomial by the highest degree term of the binomial. Write the result above the division line.
2. Multiply this result by the divisor, and subtract the resulting binomial from the polynomial.
3. Divide the highest degree term of the remaining polynomial by the highest degree term of the binomial.
4. Repeat this process until the remaining polynomial has lower degree than the binomial.

Example: Divide 2x⁴ -9x³ +21x² - 26x + 12 by 2x - 3.

The following two theorems have applications to long division:

Remainder Theorem. When a polynomial P(x) is divided by x - a, the remainder is equal to P(a).

Factor Theorem. If P(x) is a polynomial and P(a) = 0, then x - a is a factor of P(x). In other words, if the remainder when P(x) is divided by x - a is 0, then x - a is a factor of P(x).

Example: If P(x) = 3x³ -2x² + 4x - 1, use the Remainder Theorem to find the remainder when P(x) is divided by x - 2.

The remainder is 23.

Example: Is x + 3 a factor of P(x) = x⁴ +2x³ -7x² + 2x - 8?
Is x - 2 a factor of P(x) = x⁴ +2x³ -7x² + 2x - 8?

Thus x + 3 is not a factor of P(x) = x⁴ +2x³ -7x² + 2x - 8, but x - 2 is a factor of P(x).