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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) = 3x^{3} -2x^{2} + 4x - 1, use the Remainder Theorem to
find the remainder when P(x) is divided by x - 2.

P(2) = 3(2)^{3} -2(2)^{2} + 4(2) - 1 = 23.

The remainder is 23.

Example: Is x + 3 a factor of P(x) = x^{4} +2x^{3} -7x^{2} + 2x - 8?
Is x - 2 a factor of P(x) = x^{4} +2x^{3} -7x^{2} + 2x - 8?