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 an is
positive, the right arm of the graph is up.
4. If the leading coefficient an 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 anxn + an–1xn–1 + ··· + a2x2 + a1x + a0 =
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 a0 and
d is any integer that divides evenly into the leading term an.
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) = anxn + an–1xn–1 + ··· + a2x2 + a1x + a0 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.