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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:
with n a positive integer and an≠0, determines uniquely the polynomial function:
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:
Add/subtract polynomials by adding/subtracting their coefficients, respectively:
Multiply polynomials by adding all the possible products of terms:
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
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.