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This essential trains for: AMC-10, AMC-12, AIME, SAT-II, Math Kangaroo 9-10.

Definition: A set of complex numbers (coefficients) of which only a finite number (n) are non-zero:

equation

with n a positive integer and an≠0, determines uniquely the polynomial function:

equation

of degree n.

Leading coefficient: the coefficient with the largest index n such that a_n≠0.

Free term: the term that is independent of x (in the notation above this would be a0).

Degree: the largest positive integer n such that a_n ≠ 0.

Constant polynomial: any number can be thought of as a degree 0 polynomial.

Observation: A degree n≠0 polynomial has at most n+1 non-zero coefficients.

Observation:A degree n ≠ 0 polynomial has at least one non-zero coefficient.

Equality of Polynomials: Two polynomials are equal when their coefficients are respectively identical:

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equation

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Add/subtract polynomials by adding/subtracting their coefficients, respectively:

equation

where

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Multiply polynomials by adding all the possible products of terms:

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The product of a degree n polynomial and a degree m polynomial is a degree n+m polynomial.

The division of polynomials is similar to integer division in that it is a type of division with remainder. The theorem of division for polynomials is:

Division Theorem If the degree n polynomial Pn(x) is divided by the degree m polynomial Dn(x) with n ≥ m and Dm(x) ≠ 0, then there are two uniquely determined polynomials Qn-m(x) and Rk(x) and k < m such that:

equation

it is obvious that the degree of the remainder must be smaller than the degree of the divisor, otherwise division can be continued until such a condition is finally met.