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In the commutative case, the lemma is a simple consequence of a generalized form of the Cayley–Hamilton theorem, an observation made by Michael Atiyah . The special case of the noncommutative version of the lemma for right ideals appears in Nathan Jacobson ( 1945 ), and so the noncommutative Nakayama lemma is sometimes known as the Jacobson ...
This can be proved as follows. First, if r is a root of a polynomial with real coefficients, then its complex conjugate is also a root. So the non-real roots, if any, occur as pairs of complex conjugate roots. As a cubic polynomial has three roots (not necessarily distinct) by the fundamental theorem of algebra, at least one root must be real.
Multiplying this by the generating function for the complete homogeneous symmetric polynomials, one obtains the constant series 1 (equivalently, plethystic exponentials satisfy the usual properties of an exponential), and the relation between the elementary and complete homogeneous polynomials follows from comparing coefficients of t m.
The FOIL method is a special case of a more general method for multiplying algebraic expressions using the distributive law. The word FOIL was originally intended solely as a mnemonic for high-school students learning algebra. The term appears in William Betz's 1929 text Algebra for Today, where he states: [2]
As in complex analysis of functions of one variable, which is the case n = 1, the functions studied are holomorphic or complex analytic so that, locally, they are power series in the variables z i. Equivalently, they are locally uniform limits of polynomials; or locally square-integrable solutions to the n-dimensional Cauchy–Riemann equations.
Kronecker substitution is a technique named after Leopold Kronecker for determining the coefficients of an unknown polynomial by evaluating it at a single value. If p(x) is a polynomial with integer coefficients, and x is chosen to be both a power of two and larger in magnitude than any of the coefficients of p, then the coefficients of each term of can be read directly out of the binary ...
Ruffini's rule can be used when one needs the quotient of a polynomial P by a binomial of the form . (When one needs only the remainder, the polynomial remainder theorem provides a simpler method.) A typical example, where one needs the quotient, is the factorization of a polynomial p ( x ) {\displaystyle p(x)} for which one knows a root r :
At –∞ the sign of a polynomial is the sign of its leading coefficient for a polynomial of even degree, and the opposite sign for a polynomial of odd degree. In the case of a non-square-free polynomial, if neither a nor b is a multiple root of p, then V(a) − V(b) is the number of distinct real roots of P.
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