{"history":[],"article":{"id":"3421bdad-6036-4f1a-8a6b-c5155861bc11","title":"Polynomial long division","slug":"polynomial-long-division","content":":::summary\n:::\n\n# \nIn [algebra](https://bliptext.com/articles/algebra), **[polynomial](https://bliptext.com/articles/polynomial) [long division](https://bliptext.com/articles/long-division)** is an [algorithm](https://bliptext.com/articles/algorithm) for dividing a [polynomial](https://bliptext.com/articles/polynomial) by another [polynomial](https://bliptext.com/articles/polynomial) of the same or lower [degree](https://bliptext.com/articles/degree-of-a-polynomial), a generalized version of the familiar arithmetic technique called [long division](https://bliptext.com/articles/long-division).\nIt can be done easily by hand, because it separates an otherwise complex division problem into smaller ones.\nSometimes using a shorthand version called [synthetic division](https://bliptext.com/articles/synthetic-division) is faster, with less writing and fewer calculations.\nAnother abbreviated method is polynomial short division (Blomqvist's method).\nPolynomial long division is an algorithm that implements the [Euclidean division of polynomials](https://bliptext.com/articles/euclidean-division-of-polynomials), which starting from two polynomials A (the dividend) and B (the divisor) produces, if B is not zero, a [quotient](https://bliptext.com/articles/quotient) Q and a remainder R such that\nand either R = 0 or the degree of R is lower than the degree of B.\nThese conditions uniquely define Q and R, which means that Q and R do not depend on the method used to compute them.\nThe result R = 0 occurs [if and only if](https://bliptext.com/articles/if-and-only-if) the polynomial A has B as a [factor](https://bliptext.com/articles/polynomial-factorization).\nThus long division is a means for testing whether one polynomial has another as a factor, and, if it does, for factoring it out.\nFor example, if a [root](https://bliptext.com/articles/root-of-a-polynomial) r of A is known, it can be factored out by dividing A by (x – r).\n\n# Example\n\n## Polynomial long division\nFind the quotient and the remainder of the division of, the dividend, by (x-3), the divisor.\nThe dividend is first rewritten like this:\nThe quotient and remainder can then be determined as follows:\n

\nDivide 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).\nPlace the result above the bar (x3 ÷ x = x2).\n
\nMultiply the divisor by the result just obtained (the first term of the eventual quotient).\nWrite the result under the first two terms of the dividend (\nx2 · (x − 3) = x3 − 3x2\n).
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\n(\nx3 − 2x2) − (x3 − 3x2) = −2x2 + 3x2 = x2\nThen, \"bring down\" the next term from the dividend.\n
\nRepeat the previous three steps, except this time use the two terms that have just been written as the dividend.\n
\nRepeat step 4.\nThis time, there is nothing to \"bring down\".\n

\nThe polynomial above the bar is the quotient q(x), and the number left over (5) is the remainder r(x).\nThe [long division](https://bliptext.com/articles/long-division) algorithm for arithmetic is very similar to the above algorithm, in which the variable x is replaced (in base 10) by the specific number 10.\n\n## Polynomial short division\nBlomqvist's method is an abbreviated version of the long division above.\nThis pen-and-paper method uses the same algorithm as polynomial long division, but [mental calculation](https://bliptext.com/articles/mental-calculation) is used to determine remainders.\nThis requires less writing, and can therefore be a faster method once mastered.\nThe division is at first written in a similar way as long multiplication with the dividend at the top, and the divisor below it.\nThe quotient is to be written below the bar from left to right.\nDivide the first term of the dividend by the highest term of the divisor (x3 ÷ x = x2).\nPlace the result below the bar.\nx3 has been divided leaving no remainder, and can therefore be marked as used by crossing it out.\nThe result x2 is then multiplied by the second term in the divisor −3 = −3x2.\nDetermine the partial remainder by subtracting −2x2 − (−3x2) = x2.\nMark −2x2 as used and place the new remainder x2 above it.\nDivide the highest term of the remainder by the highest term of the divisor (x2 ÷ x = x).\nPlace the result (+x) below the bar.\nx2 has been divided leaving no remainder, and can therefore be marked as used.\nThe result x is then multiplied by the second term in the divisor −3 = −3x.\nDetermine the partial remainder by subtracting 0x − (−3x) = 3x.\nMark 0x as used and place the new remainder 3x above it.\nDivide the highest term of the remainder by the highest term of the divisor (3x ÷ x = 3).\nPlace the result (+3) below the bar.\n3x has been divided leaving no remainder, and can therefore be marked as used.\nThe result 3 is then multiplied by the second term in the divisor −3 = −9.\nDetermine the partial remainder by subtracting −4 − (−9) = 5.\nMark −4 as used and place the new remainder 5 above it.\nThe polynomial below the bar is the quotient q(x), and the number left over (5) is the remainder r(x).\n\n# Pseudocode\nThe algorithm can be represented in [pseudocode](https://bliptext.com/articles/pseudocode) as follows, where +, −, and × represent polynomial arithmetic, and lead(r) / lead(d) represents the polynomial obtained by dividing the two leading terms:\n**function** n / d **is**\nrequire d ≠ 0\nq ← 0\nr ← n // At each step n = d × q + r **while** r ≠ 0 **and** degree(r) ≥ degree(d) **do**\nt ← lead(r) / lead(d) // Divide the leading terms\nq ← q + t r ← r − t × d **return** (q, r)\nThis works equally well when degree(n) < degree(d); in that case the result is just the trivial (0, n).\nThis 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.\n\n# Euclidean division\nFor every pair of polynomials (A, B) such that B ≠ 0, polynomial division provides a quotient Q and a remainder R such that\nand either R=0 or degree(R) < degree(B).\nMoreover (Q, R) is the unique pair of polynomials having this property.\nThe process of getting the uniquely defined polynomials Q and R from A and B is called Euclidean division (sometimes division transformation).\nPolynomial long division is thus an [algorithm](https://bliptext.com/articles/algorithm) for Euclidean division.\n\n# Applications\n\n## Factoring polynomials\nSometimes one or more roots of a polynomial are known, perhaps having been found using the [rational root theorem](https://bliptext.com/articles/rational-root-theorem).\nIf one root r of a polynomial P(x) of degree n is known then polynomial long division can be used to factor P(x) into the form (x − r)Q(x) where Q(x) is a polynomial of degree n − 1.\nQ(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.\nLikewise, if several roots r, s,.\n. . of P(x) are known, a linear factor (x − r) can be divided out to obtain Q(x), and then (x − s) can be divided out of Q(x), etc. Alternatively, the quadratic factor can be divided out of P(x) to obtain a quotient of degree n − 2.\nThis method is especially useful for cubic polynomials, and sometimes all the roots of a higher-degree polynomial can be obtained.\nFor example, if the rational root theorem produces a single (rational) root of a [quintic polynomial](https://bliptext.com/articles/quintic-function), it can be factored out to obtain a quartic (fourth degree) quotient; the explicit formula for the roots of a [quartic polynomial](https://bliptext.com/articles/quartic-function) can then be used to find the other four roots of the quintic.\nThere is, however, no general way to solve a quintic by purely algebraic methods, see [Abel–Ruffini theorem](https://bliptext.com/articles/abel-ruffini-theorem).\n\n## Finding tangents to polynomial functions\nPolynomial long division can be used to find the equation of the line that is [tangent](https://bliptext.com/articles/tangent) to the [graph of the function](https://bliptext.com/articles/graph-of-a-function) defined by the polynomial P(x) at a particular point If R(x) is the remainder of the division of P(x) by (x – r)2, then the equation of the [tangent](https://bliptext.com/articles/tangent) line at to the [graph of the function](https://bliptext.com/articles/graph-of-a-function) is regardless of whether or not r is a root of the polynomial.\n\n### Example\nFind the equation of the line that is tangent to the following curve\nBegin by dividing the polynomial by:\nThe tangent line is y=(-21x-32)\n\n## Cyclic redundancy check\nA [cyclic redundancy check](https://bliptext.com/articles/cyclic-redundancy-check) uses the remainder of polynomial division to detect errors in transmitted messages.","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T19:08:18.823Z","updated_at":"2025-01-25T19:08:18.823Z","search_vector":"'divis':3 'long':2 'polynomi':1","search_content":" \n \n\n# \nIn [algebra](https://bliptext.com/articles/algebra), **[polynomial](https://bliptext.com/articles/polynomial) [long division](https://bliptext.com/articles/long-division)** is an [algorithm](https://bliptext.com/articles/algorithm) for dividing a [polynomial](https://bliptext.com/articles/polynomial) by another [polynomial](https://bliptext.com/articles/polynomial) of the same or lower [degree](https://bliptext.com/articles/degree-of-a-polynomial), a generalized version of the familiar arithmetic technique called [long division](https://bliptext.com/articles/long-division).\nIt can be done easily by hand, because it separates an otherwise complex division problem into smaller ones.\nSometimes using a shorthand version called [synthetic division](https://bliptext.com/articles/synthetic-division) is faster, with less writing and fewer calculations.\nAnother abbreviated method is polynomial short division (Blomqvist's method).\nPolynomial long division is an algorithm that implements the [Euclidean division of polynomials](https://bliptext.com/articles/euclidean-division-of-polynomials), which starting from two polynomials A (the dividend) and B (the divisor) produces, if B is not zero, a [quotient](https://bliptext.com/articles/quotient) Q and a remainder R such that\nand either R = 0 or the degree of R is lower than the degree of B.\nThese conditions uniquely define Q and R, which means that Q and R do not depend on the method used to compute them.\nThe result R = 0 occurs [if and only if](https://bliptext.com/articles/if-and-only-if) the polynomial A has B as a [factor](https://bliptext.com/articles/polynomial-factorization).\nThus long division is a means for testing whether one polynomial has another as a factor, and, if it does, for factoring it out.\nFor example, if a [root](https://bliptext.com/articles/root-of-a-polynomial) r of A is known, it can be factored out by dividing A by (x – r).\n\n# Example\n\n## Polynomial long division\nFind the quotient and the remainder of the division of, the dividend, by (x-3), the divisor.\nThe dividend is first rewritten like this:\nThe quotient and remainder can then be determined as follows:\n \n \nDivide 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).\nPlace the result above the bar (x3 ÷ x = x2).\n \n \nMultiply the divisor by the result just obtained (the first term of the eventual quotient).\nWrite the result under the first two terms of the dividend (\nx2 · (x − 3) = x3 − 3x2\n). 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\n(\nx3 − 2x2) − (x3 − 3x2) = −2x2 + 3x2 = x2\nThen, \"bring down\" the next term from the dividend.\n \n \nRepeat the previous three steps, except this time use the two terms that have just been written as the dividend.\n \n \nRepeat step 4.\nThis time, there is nothing to \"bring down\".\n \n \nThe polynomial above the bar is the quotient q(x), and the number left over (5) is the remainder r(x).\nThe [long division](https://bliptext.com/articles/long-division) algorithm for arithmetic is very similar to the above algorithm, in which the variable x is replaced (in base 10) by the specific number 10.\n\n## Polynomial short division\nBlomqvist's method is an abbreviated version of the long division above.\nThis pen-and-paper method uses the same algorithm as polynomial long division, but [mental calculation](https://bliptext.com/articles/mental-calculation) is used to determine remainders.\nThis requires less writing, and can therefore be a faster method once mastered.\nThe division is at first written in a similar way as long multiplication with the dividend at the top, and the divisor below it.\nThe quotient is to be written below the bar from left to right.\nDivide the first term of the dividend by the highest term of the divisor (x3 ÷ x = x2).\nPlace the result below the bar.\nx3 has been divided leaving no remainder, and can therefore be marked as used by crossing it out.\nThe result x2 is then multiplied by the second term in the divisor −3 = −3x2.\nDetermine the partial remainder by subtracting −2x2 − (−3x2) = x2.\nMark −2x2 as used and place the new remainder x2 above it.\nDivide the highest term of the remainder by the highest term of the divisor (x2 ÷ x = x).\nPlace the result (+x) below the bar.\nx2 has been divided leaving no remainder, and can therefore be marked as used.\nThe result x is then multiplied by the second term in the divisor −3 = −3x.\nDetermine the partial remainder by subtracting 0x − (−3x) = 3x.\nMark 0x as used and place the new remainder 3x above it.\nDivide the highest term of the remainder by the highest term of the divisor (3x ÷ x = 3).\nPlace the result (+3) below the bar.\n3x has been divided leaving no remainder, and can therefore be marked as used.\nThe result 3 is then multiplied by the second term in the divisor −3 = −9.\nDetermine the partial remainder by subtracting −4 − (−9) = 5.\nMark −4 as used and place the new remainder 5 above it.\nThe polynomial below the bar is the quotient q(x), and the number left over (5) is the remainder r(x).\n\n# Pseudocode\nThe algorithm can be represented in [pseudocode](https://bliptext.com/articles/pseudocode) as follows, where +, −, and × represent polynomial arithmetic, and lead(r) / lead(d) represents the polynomial obtained by dividing the two leading terms:\n**function** n / d **is**\nrequire d ≠ 0\nq ← 0\nr ← n // At each step n = d × q + r **while** r ≠ 0 **and** degree(r) ≥ degree(d) **do**\nt ← lead(r) / lead(d) // Divide the leading terms\nq ← q + t r ← r − t × d **return** (q, r)\nThis works equally well when degree(n) < degree(d); in that case the result is just the trivial (0, n).\nThis 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.\n\n# Euclidean division\nFor every pair of polynomials (A, B) such that B ≠ 0, polynomial division provides a quotient Q and a remainder R such that\nand either R=0 or degree(R) < degree(B).\nMoreover (Q, R) is the unique pair of polynomials having this property.\nThe process of getting the uniquely defined polynomials Q and R from A and B is called Euclidean division (sometimes division transformation).\nPolynomial long division is thus an [algorithm](https://bliptext.com/articles/algorithm) for Euclidean division.\n\n# Applications\n\n## Factoring polynomials\nSometimes one or more roots of a polynomial are known, perhaps having been found using the [rational root theorem](https://bliptext.com/articles/rational-root-theorem).\nIf one root r of a polynomial P(x) of degree n is known then polynomial long division can be used to factor P(x) into the form (x − r)Q(x) where Q(x) is a polynomial of degree n − 1.\nQ(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.\nLikewise, if several roots r, s,.\n. . of P(x) are known, a linear factor (x − r) can be divided out to obtain Q(x), and then (x − s) can be divided out of Q(x), etc. Alternatively, the quadratic factor can be divided out of P(x) to obtain a quotient of degree n − 2.\nThis method is especially useful for cubic polynomials, and sometimes all the roots of a higher-degree polynomial can be obtained.\nFor example, if the rational root theorem produces a single (rational) root of a [quintic polynomial](https://bliptext.com/articles/quintic-function), it can be factored out to obtain a quartic (fourth degree) quotient; the explicit formula for the roots of a [quartic polynomial](https://bliptext.com/articles/quartic-function) can then be used to find the other four roots of the quintic.\nThere is, however, no general way to solve a quintic by purely algebraic methods, see [Abel–Ruffini theorem](https://bliptext.com/articles/abel-ruffini-theorem).\n\n## Finding tangents to polynomial functions\nPolynomial long division can be used to find the equation of the line that is [tangent](https://bliptext.com/articles/tangent) to the [graph of the function](https://bliptext.com/articles/graph-of-a-function) defined by the polynomial P(x) at a particular point If R(x) is the remainder of the division of P(x) by (x – r)2, then the equation of the [tangent](https://bliptext.com/articles/tangent) line at to the [graph of the function](https://bliptext.com/articles/graph-of-a-function) is regardless of whether or not r is a root of the polynomial.\n\n### Example\nFind the equation of the line that is tangent to the following curve\nBegin by dividing the polynomial by:\nThe tangent line is y=(-21x-32)\n\n## Cyclic redundancy check\nA [cyclic redundancy check](https://bliptext.com/articles/cyclic-redundancy-check) uses the remainder of polynomial division to detect errors in transmitted messages.","revisionCount":0},"hasMore":false,"nextCursor":null}

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