To divide by a nonzero polynomial over a field, subtract the multiple of that cancels the leading term of the current remainder, and add that multiple to the quotient. Each step strictly decreases the remainder degree, so the process terminates with and . If two such pairs existed, would have degree both at least and less than , unless both differences vanished.
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Polynomial long division is a method used to divide one polynomial by another polynomial, similar to the long division process used with numbers. It involves a systematic way of dividing polynomials, which results in a quotient and, in some cases, a remainder.