You should KNOW

1. That a polynomial in x is a sum of terms, each of which is a number (coefficient) multiplied by a non-negative integer power of x, that the highest power that appears is the degree of the polymomial, and that the term with highest power is called the leading term.

2. How to add, subtract and multiply polynomials, most importantly, how to multiply two polynomials each of which has more than two terms. (The so-called FOIL method does not work in this case and should generally be avoided.)

3. How to perform long division of one polynomial P(x) by another D(x) to get a polynomial quotient Q(x) and remainder R(x) (with degree smaller than that of D(x)), and that then
P(x) = Q(x)D(x) + R(x).
(You do NOT need to know "synthetic division.")

4. The Remainder Theorem, which says that when P(x) is divided by x-c, the remainder is R(x) = c.

5. The Factor Theorem, which says that if x-c is a factor then c is a zero and vice versa, and how this can be used sometimes to aid factorization. (Note: A polynomial with complex zeros will have factors that involve complex numbers. Thus the polynomial P(x) = x2 + 4 factors into (x - 2i) (x + 2i) if we allow complex numbers, but does not factor at all if we allow only real numbers.)

6. What is meant by the multiplicity of a root or zero, and that a polynomial of degree n has n possibly complex roots counting multiplicity.

You should be ABLE TO

1. Add, subtract and multiply polynomials and simplify results, especially in cases where the FOIL method is not applicable.

2. Divide one polynomial by another to get a polynomial remainder and quotient and to explain how these are related.

3. Find remainders when dividing by x-c by using the Remainder Theorem.

4. Use the Factor Theorem to facilitate factoring when a zero can be identified and use this to solve simple polynomial equations.

5. Find a polynomial that has given roots.