Polynomial Facts

Facts about polynomials of the form p(x) = a_nxⁿ + a_n–1x^n–1 + ··· + a₂x² + a₁x + a₀ are listed below.

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_nxⁿ + a_n–1x^n–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 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_nxⁿ + a_n–1x^n–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.