What is Polynomial division: Definition and 39 Discussions

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).

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  1. C

    Simple Induction Induction proof of Polynomial Division Theorem

    Theorem: Let ## f(x), g(x) \in \mathbb{F}[ x] ## by polynomials, s.t. the degree of ## g(x) ## is at least ## 1 ##. Then: there are polynomials ## q(x), r(x) \in \mathbb{F}[ x] ## s.t. 1. ## f(x)=q(x) \cdot g(x)+r(x) ## or 2. the degree of ## r(x) ## is less than the degree of ## g(x) ## Proof...
  2. S

    Remainder of polynomial division

    ##x^{2017} + 1 = Q(x) . (x-1)^2 + ax + b## where ##Q(x)## is the quotient and ##ax+b## is the remainder ##x=1 \rightarrow 2 =a+b## Then how to proceed? Thanks
  3. Euler2718

    I Finding a non-convergent Cauchy sequence

    Define a metric on ##\mathbb{R}[x]## for distinct polynomials ##f(x),g(x)## as ##d(f(x),g(x)) = \frac{1}{2^{n}}##, where ##n## is the largest positive integer such that ##x^{n}## divides ##f(x)-g(x)##. Equivalently, ##n## is the multiplicity of the root ##x=0## of ##f(x)-g(x)##. Set...
  4. Adgorn

    B Questions regarding polynomial divisions and their roots

    Hello everyone, Going through calculus study, there is a vague point regarding polynomials I'd like to make clear. Say there's a polynomial ##f## with a root at ##a## with multiplicity ##2##, i.e. ##f(x)=(x-a)^2g(x)## where ##g## is some other polynomial. I define ##h(x)=\frac {f(x)} {x-a}##...
  5. C

    Factoring Combinatorial Functions

    Homework Statement Define {x \choose n}=\frac{x(x-1)(x-2)...(x-n+1)}{n!} for positive integer n. For what values of positive integers n and m is g(x)={{{x+1} \choose n} \choose {m}}-{{{x} \choose n} \choose {m}} a factor of f(x)={{{x+1} \choose n} \choose {m}}? Homework Equations The idea...
  6. S

    B Remainder of polynomial division

    Is this true? If the remainder of f(x) / g(x) is a (where a is constant), then the remainder of (f(x))n / g(x) is an I don't know how to be sure whether it is correct or wrong. I just did several examples and it works. Thanks
  7. H

    5 degree equation contains imaginary value?

    Homework Statement Homework Equations polynomial equation $$(a+b)^2 = \sum_{k=0}^{n} \binom nk a^{n-k} b^k $$ The Attempt at a Solution i get $$ \frac {1} {32} + \frac {5} {16}\frac {3^{0.5}i} {2} +\frac {10} {8}\frac {3i^2} {4} + \frac {10} {4}\frac {3^{1.5}i^3} {8} + \frac {5} {2} \frac...
  8. M

    I Polynomial Division: Solve with Mike's Help!

    Hello everyone. Iam working on a course in digital control systems and by reading my textbook I stumbled over this expression. C(z) = 0.3678z + 0.2644 : z^2 − 1.3678z + 0.3678 = 0.3678z^−1 + 0.7675z^−2 + 0.9145z^−3 + ... Now Iam wondering how the result of the polynomial division is...
  9. kaliprasad

    MHB Polynomial Division: Finding Q(x) for P(x)$x^3$

    Let P(x) be a polynomial of x. Show that there exists a polynomial Q(x) such that P(x)Q(x) is a polynomial of $x^3$
  10. M

    MHB How can I divide a polynomial by (x+k) using synthetic division?

    i am trying to divide x^3+(1-k^2)x+k by (x+k) but i can't do this can you show me how to.
  11. stungheld

    Finding Solutions for Polynomial Division: Where to Begin?

    Homework Statement How many pairs of solutions make x^4 + px^2 + q = 0 divisable by x^2 + px + q = 0 Homework Equations x1 + x2 = -p x1*x2= q[/B] The Attempt at a Solution I tried making z = x^2 and replacing but got nowhere. I figure 0,1,-1 are 3 numbers that fit but I am not sure what's...
  12. J

    Polynomial Division (continued from osnarf's problem)

    Hello, My problem is the same as osnarf's problem in thread "Polynomial division proof", https://www.physicsforums.com/threads/polynomial-division-proof.451991/ But, I would like some further help. The problem: Prove that for any polynomial function f, and any number a, there is a polynomial...
  13. C

    Dividing Polynomials: How to Solve (4s^3+4s^2+72)/(s+3)

    Mod note: Moved from a technical math section, so missing the template. I have this question and the answer but my mathXL does not show me how it came to this conclusion. (4s3+4s2 + 72)/ s+3I got all the way to the answer 4s2 - 8s The correct answer is 4s2 - 8s + 24 I just don't know the...
  14. J

    I don't understand polynomial division

    At first he shows 2x+4 / 2 and you just divide both 2x and 4 by 2. But then in the next example he is dividing x^2+3x+6 by x+1 and he doesn't divide x^2 by x+1, 3x by x+1 and 6 by x+1. I do not understand how he does the problem.
  15. M

    Inverse laplace transform (polynomial division? Complex roots?)

    Homework Statement Decide the inverse laplace transform of the problem below: F(s)= \frac{4s-5}{s^2-4s+8} You're allowed to use s shifting. Homework Equations The Attempt at a Solution By looking at the denominator, I see that it might be factorized easily, so I try that...
  16. MarkFL

    MHB Ray's question at Yahoo Answers regarding polynomial division

    Here is the question: I have posted a link there to this topic so the OP can see my work.
  17. C

    Polynomial Division: Simplifying with Long Division

    1. Hi I have a question I am stuck on it is: (x3 + 2x2y - 2xy2 - y3)/(x-y) Can anyone help? Homework Equations The Attempt at a Solution
  18. E

    Polynomial Division 2: Step-by-Step Solution for (x^3-3x^2+12x-5)/(x-2)"

    Homework Statement Show by polynomial division that \frac{x^3-3x^2+12x-5}{x-2}=(x^2-x+10)+\frac{15}{x-2} The Attempt at a Solution Please see attachment
  19. E

    P.S. I did it on paperPolynomial Division: 3x^3-5x^2+10x+4 / 3x+1

    Homework Statement Use polynomial long division to determine the quotient when 3x^3-5x^2+10x+4 divided by 3x+1 The Attempt at a Solution Please see attachment as I wasn't quite sure how to write my answer here :shy:
  20. MarkFL

    MHB Gabriel's question at Yahoo Answers regarding polynomial division and remainders

    Here is the question: Here is a link to the question: The remainder of f(x)/(x^2+x+1) and f(x)/[(x+1)^2] are x+5 and x-1 respectively.? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  21. S

    Polynomial Division Homework: Q[x], Z5[x], R[x]

    Homework Statement Divide 2x5+x-1 by 3x2+1 in Q[x], Z5[x], and R[x] The Attempt at a Solution I believe the answer should be the same in Q[x] and R[x] and after division I got 2/3x3-2/9x with remainder 1 2/9x-1. I had trouble in Z5 and would like someone to help me. So far I...
  22. X

    Polynomial Division: Sum of Remainders for 53 < k < 115

    Homework Statement This question was on a test in a math contest I was recently in, and I cannot seem to figure out how to get the answer: Let f(x)=x7+x6+x5+x4+x3+x2+x+1. If k is a positive integer such that 53 < k < 115, find the sum of all distinct k such that the numerical remainder...
  23. S

    What is the Remainder of Polynomial Division in Z5[x] by x+3?

    Homework Statement Find the remainder of x^4 + 3x +2 after division by x+3 in Z5[x] Homework Equations my quotient after dividing was: x^3 + 2X^2 + 4x +1 The Attempt at a Solution I found the remainder to be 4. If anyone has time, I believe I made a mistake somewhere and would...
  24. O

    How Does Polynomial Long Division Validate the Existence of Remainders?

    Spivak's "Calculus," chapter 3 - problem 7 - a Homework Statement Prove that for any polynomial function f, and any number a, there is a polynomial function g, anad a number b, such that f(x) = (x - a)*g(x) + b for all x. (The idea is simply to divide (x - a) into f(x) by long division, until...
  25. C

    Polynomial Division: Show g(x) Divides f(x)

    Homework Statement Show that g(x) = x^3 + 1 divides f(x) = x^{9999} +1. Homework Equations The Attempt at a Solution g(x) can obviously be factored into the irreducible polynomials (x+1)(x^2 - x + 1) in Z[x], and since f(-1) = (-1)^{9999} + 1 = 0, the factor theorem gives that (x+1) divides...
  26. M

    Simplification of polynomial division

    Homework Statement For all x>2, \frac{2x²+2x-12}{x-2} simplifies to: 2(x - 2), x + 3, 2(x + 3)(x - 2), x - 2 or 2(x + 3). (Problem taken from http://www.analyzemath.com/practice_tests/act/act_sample_1.html" , question five). Homework Equations None, AFAIK. The Attempt at a...
  27. H

    Polynomial Division: Simplifying 3rd Roots in Denominator

    3rd roots in denominator Homework Statement \frac{ \sqrt[3]{25} + \sqrt[3]{5x} + \sqrt[3]{x^2} }{ \sqrt[3]{x} - \sqrt[3]{5} } Rewrite the expression with no roots in the denominator and it being simplified as far as possible.Homework Equations The Attempt at a Solution I'm.. stumped...
  28. M

    What is the Correct Quotient and Remainder When Dividing this Polynomial?

    Here is the problem I am working on: Find the quotient and remainder when P(x) = 7 x^6 - 9 x^5 + 8 x^4 + 9 x^3 + 4 x^2 - 6 x is divided by (x + 5). My answer that I came up with is this. Q = 7x^5 - 44x^4 + 228x^3 - 1131x^2 + 5659x R = -28301x I have done this using Long and Synthetic...
  29. H

    Solve Polynomial Division: Find K Given Remainder of 3

    Homework Statement The remainder is 3 when 4x^2 + 9x + k is divided by x-1 Im suppose to find the K. I am just confused because this question they give me the remainder..so what am i suppose to do with it?
  30. D

    Find Factors of Polynomial Division: x-2 in Q[x] & x+1 in Z5[x]

    1. Polynomial division a) For what values of k is x-2 a factor x^4 – 5x^3 + 3x + k in Q[x]? b) For what values of k is x+1 a factor of x^4 + 2x^3 – 3x^2 + kx + 1 in Z5[x]
  31. S

    How can I solve this advanced polynomial division problem?

    Some of you may have read my troubles that I had in this thread https://www.physicsforums.com/showthread.php?t=162079 Now I am having more troubles with more advanced problems. I started with this problem: [4y^2 - 7y - 12] / [(y)(y+2)(y-3)] The problem is that I set it up the same...
  32. S

    Polynomial Division: Solving Denominator > Numerator

    I just can't remember how to do this! I've been to several sites suggesting synthetic division and other guides on polynomial division but i can't get it into my head and its driving me wild. (8x-8)/(x^2+3x+2) synthetic division doesn't work here because the denominator is larger than the...
  33. S

    Partial fraction expansion (polynomial division)

    If we're asked to expand an irrational partial fraction, we need to rationalize it first, right? I've forgotten (well not totally) how to perform polynomial division. Here are a few examples: 1. (1000x +1000000)/(.4x + 200). For this one I got 50000. 2. (500x + 60000)/(x+100)...
  34. R

    Polynomial Division puzzler(well to me anyway :P )

    O.K I factored and reduced an equation I am doing down to (s+5t)/(t-s) I could not figure it out so I went to webmath and used a calculator to find out the answer so I could reverse solve the problem. The answer is 1 according to webmath. Any hints on how to get started on this becuase I...
  35. H

    HighSchool Calculus Polynomial Division Help

    I was just doing my homework when I stumbled on this question. I don't understand it and I was wondering if someone could help me through it (don't just do it though, because I want to learn HOW to do it :smile: ) When 8x^3+4x^2-px+6, p E R, is divided by 2x-1, the remainer is 3. Determine...
  36. H

    Polynomial Division Help - x^3 + 4x^2 - 3x - 12 & -x^3 + 75x - 250

    I can't seem to get it... I need some help with 2 of the problems below: x^3 + 4x^2 - 3x - 12 over x^2 - 3 and -x^3 + 75x - 250 over x + 10 I've tried tutorials and everything, I can get a few but these are just a little tricky for me. Any help is appreciated. :)
  37. M

    Solving a Polynomial Division Problem: Remainder = 5 & 7

    Given that a polymial p(x) is p(x)= (x-1)(x-2) q(x) + 2x+3 where q(x) is also a polynomial Find the remainder when p(x) is divided by (x-1)(x+2) where the remainder divided by (x-1) and (x+2) is both 5 and 7 respectively. I don't know even where to start ! so please help, thanks alot.
  38. A

    Solve Polynomial Division: -6x^4+2x^2-8x+1 / 2x^2-3

    I have been trying two problems for the longest time, and no one is able to help me I am so stuck, I know how to divide polynomials using long division and synthetic division using simpler numbers but I just can't do these two questions and my course is online, so I can't even ask my teacher...
  39. S

    Problem with polynomial division and 'i'

    I was learning polynomial division, and I can do most problems, except this one which is bothering me. : 3x^2 + 2x + 7 --------------- (1+i)x - 2 How would I divide something like that? Nothing is working. Thanks.