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Algebra II: Polynomials

Terms

Introduction and Summary

Nested Form of a Polynomial

Conjugate Zeros Theorem  -  If P(x) is a polynomial with real coefficients, and if a + bi is a zero of P , then a - bi is a zero of P .
Factor Theorem  -  If P(x) is a polynomial and P(a) = 0 , then x - a is a factor of P(x) . In other words, if the remainder when P(x) is divided by x - a is 0, then x - a is a factor of P(x) .
Fundamental Theorem of Algebra  -  Every polynomial function of positive degree with complex coefficients has at least one complex zero.
Corollary. Every polynomial function of positive degree n has exactly n complex zeros (counting multiplicities).
Multiplicity  -  A function with n identical roots is said to have a zero of multiplicity n .
Nested Form  -  The form of a polynomial P(x) = (((((a)x + b)x + c)x + d )x + ... ) .
Rational Zeros Theorem  -  If P(x) is a polynomial with integer coefficients and if is a zero of P(x) (if P() = 0 ), then p is a factor of the constant term of P(x) and q is a factor of the leading coefficient of P(x) .
Remainder Theorem  -  When a polynomial P(x) is divided by x - a , the remainder is equal to P(a) .
Root  -  A number that, when plugged in for the variable, sets a function equal to zero. Also called a zero.
Synthetic Division  -  A process by which a polynomial is divided by a binomial, in which the coefficients of the polynomial are placed in a row and multiplied by and added to the constant divisor as in nested form.
Zero  -  A number that, when plugged in for the variable, sets a function equal to zero. Also called a root.

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