Section 11
Polynomials
Divides polynomials with quotient and remainder.
The remainder when dividing by x-a is P(a).
Relates roots and linear factors.
Complete factorization over a field that contains the roots.
Relates the roots and coefficients of a quadratic.
Narrows the rational candidates for roots of an integer polynomial.
Generalizes single-variable polynomials to expressions in several algebraic variables.
where is a multi-index of nonnegative integers.
Equivalent form
Equivalent form
Measures the degree of multivariable terms by summing their exponents.
its total degree is
The total degree of a nonzero polynomial is
Example
Equivalent form
Every term has the same total degree.
is homogeneous of degree .
A characteristic property is
Finds points that simultaneously satisfy several polynomials.
For two univariate polynomials and , the resultant can be defined via the Sylvester matrix
Over a field, and considering roots in an algebraic closure
Algebraically detects whether two polynomials share a root and serves as an elimination tool.