Polynomial division Definition and 39 Threads

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...

##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

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...

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}##...

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...

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

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...

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...

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 $x^3$

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.

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...

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...

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...

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.

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...

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

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

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

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:

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.

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...

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...

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...

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...

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...

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...

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...

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...

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?

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]

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...

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...

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)...

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...

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...

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. :)

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.

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...

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.