Polynomial End Behavior:

1. If the degree *n* of a polynomial is even, then the
arms of the graph are either both up or both down.

2. If the degree *n* is odd, then one arm of the graph
is up and one is down.

3. If the leading coefficient *a*_{n} is
positive, the right arm of the graph is up.

4. If the leading coefficient *a*_{n} is
negative, the right arm of the graph is down.

Extreme Values:

The graph of a polynomial of degree *n* has
at most *n* – 1 extreme
values.

Inflection Points:

The graph of a polynomial of degree *n* has
at most *n* – 2 inflection points.

Remainder Theorem:

*p*(*c*) is the remainder when
polynomial *p*(*x*)
is divided by *x* – *c*.

Factor Theorem:

*x* – *c* is a factor of
polynomial *p*(*x*) if and only if *c*
is a zero of *p*(*x*).

Rational Root Theorem:

If a polynomial equation *a*_{n}x^{n} + *a*_{n}_{–1}*x*^{n}^{–1} + ··· + *a*_{2}*x*^{2} + *a*_{1}*x* + *a*_{0} =
0 has
integer coefficients then it is possible to make a complete list
of all possible rational roots.
This list consists of all possible numbers of the form *c*/*d*, where *c* is any integer
that divides evenly into the constant term *a*_{0} and
*d* is any integer that divides evenly into the leading term *a*_{n}.

Conjugate Pair Theorem:

If a polynomial has real coefficients then
any complex zeros occur
in complex conjugate pairs. That is, if *a* + *bi* is
a zero then so is *a* – *bi*, where *a* and *b* are
real numbers.

Fundamental Theorem
of Algebra:

A polynomial *p*(*x*) = *a*_{n}x^{n} + *a*_{n}_{–1}*x*^{n}^{–1} + ··· + *a*_{2}*x*^{2} + *a*_{1}*x* + *a*_{0} of
degree at least 1 and with coefficients that may be real or
complex must have a factor of the form *x* – *r*,
where *r* may be real or complex.

Corollary of the Fundamental Theorem of Algebra:

A polynomial of degree *n* must have exactly *n* zeros,
counting mulitplicity.