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 Polynomial Facts Facts about polynomials of the form p(x) = anxn + an–1xn–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 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.   See also