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Concept
Polynomial long division
Summary
In algebra, polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree, a generalized version of the familiar arithmetic technique called long division. It can be done easily by hand, because it separates an otherwise complex division problem into smaller ones. Sometimes using a shorthand version called synthetic division is faster, with less writing and fewer calculations. Another abbreviated method is polynomial short division (Blomqvist's method).
Polynomial long division is an algorithm that implements the Euclidean division of polynomials, which starting from two polynomials A (the dividend) and B (the divisor) produces, if B is not zero, a quotient Q and a remainder R such that
A = BQ + R,
and either R = 0 or the degree of R is lower than the degree of B. These conditions uniquely define Q and R, which means that Q and R do not depend on the method used to compute them.
The result R = 0 occurs if and only if the polynomial A has B as a factor. Thus long division is a means for testing whether one polynomial has another as a factor, and, if it does, for factoring it out. For example, if a root r of A is known, it can be factored out by dividing A by (x – r).
Find the quotient and the remainder of the division of the dividend, by the divisor.
The dividend is first rewritten like this:
The quotient and remainder can then be determined as follows:
Divide the first term of the dividend by the highest term of the divisor (meaning the one with the highest power of x, which in this case is x). Place the result above the bar (x3 ÷ x = x2).
Multiply the divisor by the result just obtained (the first term of the eventual quotient). Write the result under the first two terms of the dividend (x2 · (x − 3) = x3 − 3x2).
Subtract the product just obtained from the appropriate terms of the original dividend (being careful that subtracting something having a minus sign is equivalent to adding something having a plus sign), and write the result underneath ((x3 − 2x2) − (x3 − 3x2) = −2x2 + 3x2 = x2).
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