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Additional info for Polynomial identities in ring theory, Volume 84 (Pure and Applied Mathematics)

Example text

3. For which values of a and b does the polynomial x 3 + ax + b have a multiple root? 4. Prove that the polynomial x n + ax m + b, n > m, does not have nonzero roots of multiplicity three or more. 5. ,,(x). Find a formula for (/112)" analogous to formula (18) (but, of course, somewhat more complicated). 6. Prove that the derivative of a polynomial is identical to zero if and only if the polynomial is a constant (that is, it has the degree zero). 7. Prove that for any polynomial f(x), there exists a polynomial g(x) such that g'(x) = f(x) and that all such polynomials g(x) (for a given f(x)) differ only in their free term.

What does it mean that the equation f(x) = 0 has two equal roots? We can write any root of an equation on paper as many times as we like, and they will always be equal numbers! But in the case of a quadratic equation, we appealed to the formula for its solution for an answer. And in the general case, we want to use some additional considerations to give a reasonable definition of when we must think that the equation f(x) = 0 has two equal roots x = a and x = a. The Bezout theorem (Theorem 13) suggests such considerations.

We begin with some remarks about calculating the sum in the general form. Let ao, aI, a2,"" an, ... be an infinite sequence of numbers. We are interested in the sums of its consecutive members: ao, ao + aI, aO + al + a2, ... , aO + al + a2 + ... + an' .... The first sequence is denoted by the letter a, and its (n+ 1)th member is then an (it is more convenient to denote the (n+1)th member thus, and not the nth, beginning the sequence with ao). The associated sequence of sums is denoted by Sa, and its (n+1)th member is then equal to (Sa)n = ao + al + a2 + ...

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Polynomial identities in ring theory, Volume 84 (Pure and Applied Mathematics) by Author Unknown

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