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).
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
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...
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...
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.