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.
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 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 dividend is first rewritten like this: The quotient and remainder can then be determined as follows: The polynomial above the bar is the quotient q(x), and the number left over (5) is the remainder r(x).
Blomqvist's method[1] is an abbreviated version of the long division above.
This pen-and-paper method uses the same algorithm as polynomial long division, but mental calculation is used to determine remainders.
This requires less writing, and can therefore be a faster method once mastered.
The division is at first written in a similar way as long multiplication with the dividend at the top, and the divisor below it.
x3 has been divided leaving no remainder, and can therefore be marked as used by crossing it out.
The polynomial below the bar is the quotient q(x), and the number left over (5) is the remainder r(x).
The algorithm can be represented in pseudocode as follows, where +, −, and × represent polynomial arithmetic, and lead(r) / lead(d) represents the polynomial obtained by dividing the two leading terms: This works equally well when degree(n) < degree(d); in that case the result is just the trivial (0, n).
This algorithm describes exactly the above paper and pencil method: d is written on the left of the ")"; q is written, term after term, above the horizontal line, the last term being the value of t; the region under the horizontal line is used to compute and write down the successive values of r.
Q(x) is simply the quotient obtained from the division process; since r is known to be a root of P(x), it is known that the remainder must be zero.
There is, however, no general way to solve a quintic by purely algebraic methods, see Abel–Ruffini theorem.
Polynomial long division can be used to find the equation of the line that is tangent to the graph of the function defined by the polynomial P(x) at a particular point x = r.[3] If R(x) is the remainder of the division of P(x) by (x – r)2, then the equation of the tangent line at x = r to the graph of the function y = P(x) is y = R(x), regardless of whether or not r is a root of the polynomial.
Find the equation of the line that is tangent to the following curve
A cyclic redundancy check uses the remainder of polynomial division to detect errors in transmitted messages.