**Polynomial Facts**

Facts about polynomials of the form *p*(*x*) = *a _{n}x^{n}* +

Polynomial End Behavior:

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

2. If the degreenis odd, then one arm of the graph is up and one is down.

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

4. If the leading coefficientais negative, the right arm of the graph is down._{n}Extreme Values:

The graph of a polynomial of degreenhas at mostn– 1 extreme values.Inflection Points:

The graph of a polynomial of degreenhas at mostn– 2 inflection points.Remainder Theorem:

p(c) is the remainder when polynomialp(x) is divided byx–c.Factor Theorem:

x–cis a factor of polynomialp(x) if and only ifcis a zero ofp(x).Rational Root Theorem:

If a polynomial equationa+_{n}x^{n}a_{n}_{–1}x^{n}^{–1}+ ··· +a_{2}x^{2}+a_{1}x+a_{0}= 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 formc/d, wherecis any integer that divides evenly into the constant terma_{0}anddis any integer that divides evenly into the leading terma._{n}Conjugate Pair Theorem:

If a polynomial has realcoefficients then any complexzeros occur in complex conjugate pairs. That is, ifa+biis a zero then so isa–bi, whereaandbare real numbers.Fundamental Theorem of Algebra:

A polynomialp(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 formx–r, wherermay be real or complex.Corollary of the Fundamental Theorem of Algebra:

A polynomial of degreenmust have exactlynzeros, counting mulitplicity.

**See also**

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