# polynomial facts

Polynomial Facts

Facts about polynomials of the form p(x) = anxn + an1xn1 + ··· + a2x2 + a1x + a0 are listed below.

Polynomial End Behavior:
1. If the degreen 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 coefficientan 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 degreen has at most n – 1 extreme values.

Inflection Points:
The graph of a polynomial of degreen has at most n – 2 inflection points.

Remainder Theorem:
p(c) is the remainder when polynomialp(x) is divided by xc.

Factor Theorem:
xc is a factor of polynomialp(x) if and only ifc is a zero of p(x).

Rational Root Theorem:
If a polynomial equation anxn + an1xn1 + ··· + a2x2 + a1x + a0 = 0 has integercoefficients then it is possible to make a complete list of all possible rationalroots.This list consists of all possible numbers of the form c/d, where c is any integer that divides evenly into the constant terma0 and d is any integer that divides evenly into the leading terman.

Conjugate Pair Theorem:
If a polynomial has realcoefficients then any complexzeros occur in complex conjugate pairs. That is, if a + bi is a zero then so is abi, where a and b are real numbers.

Fundamental Theorem of Algebra:
A polynomialp(x) = anxn + an1xn1 + ··· + a2x2 + a1x + a0 of degree at least 1 and with coefficients that may be real or complex must have a factor of the form xr, where r may be real or complex.

Corollary of the Fundamental Theorem of Algebra:
A polynomial of degreen must have exactly nzeros, counting mulitplicity.