2
votes
1answer
42 views

Silly number theory questions I can't prove.

I know if $gcd(r,s)=1$ then $1=as+bs$ for some intgers $a,b$. Here's what I want to know: which numbers can be written as $as+bs$, if I am restricted to $a,b \in \mathbb{N}$? To be more specific, I ...
-1
votes
2answers
25 views

Multiplicity of roots in finite fields of order prime. [closed]

I am having trouble with completing this question from last years exam (part a and d) Let p be a prime, and $f = x^5-1 = (X-1)(X^4+X^3+X^2+X+1) \in \mathbb{F}_p[X]$ Show: (a) if $p\neq 5$ then every ...
0
votes
2answers
26 views

Equation Theory: A polynomial with specific remainders when divided by specific divisors. What is the remainder when divided by BOTH divisors

Again, for my Equation Theory class, I have the subject question.$p(x)$ has a remainder of 3 when divided by $x-1$ and a remainder of 5 when divided by $x-3$. What is the remainder when $p(x)$ is ...
0
votes
1answer
43 views

Polynomials - getting wrong answer using Euclidean algorithm

I am finding the GCD of $a = x^3 + 11/3x^2 + 17/4x + 3/2$ and $b = 3x^2 + 22/3x + 17/4$ using the Euclidean algorithm. So I divide $a/b$ and get $q$ and $r$ such that $a = qb + r$. Then, according to ...
1
vote
2answers
50 views

Question about kth root of a reduced ring element.

Let $n > 1$ be a positive integer. Let $k > 1$ be a positive integer. Define the reduced polynomial rings $f_n = \Bbb R[X_n]/(1+(X_n)^{n})$ How do we know if $(X_n)^{1/k}$ is an element of ...
0
votes
1answer
77 views

Have you seen this theorem before? (GCD divides, neccessary & sufficient condition)

Conjecture. Let $a,b, c\in \Bbb{Z}, b \neq 0$, The following conditions are equivalent: (1) $d = \gcd(a,b)$ divides c. (2) There's a polynomial in $f \in \Bbb{Z}[X,Y]$ with $c$ constant term, such ...
0
votes
0answers
31 views

Let $f(x)\in K[x]$ ($K$ field). Prove that if $(f(x),f´(x))=1$ (greatest common divisor is 1) then $f(x)$ does not have multiple roots in $K$

Please I would like you to tell me if my proof is correct Let $f(x)\in K[x]$ ($K$ field). Prove that if $(f(x),f´(x))=1$ (greates common divisor is 1) then $f(x)$ does not have multiple roots in $K$ ...
0
votes
1answer
26 views

$(x+b)^3\mid P(x)+a$ and $(x-a)^3\mid P(x)-a$

$a,b\in\mathbb{C}$, $b!=0$ I need to find all the polynomials $P$ of degree $5$ verifying: $ \begin{cases}(x+b)^3\mid P(x)+a\\ (x-b)^3\mid P(x)-a\end{cases} $ PS : there was en error, i fixed it ...
1
vote
0answers
34 views

GCD of this polynomial

So here is the exact question that i am having trouble on: "Extend the Euclidean algorithm to polynomials and find the greatest common divisor of: $3x^5-10x^4-4x^3-14x^2-7x-4$ and ...
3
votes
3answers
234 views

A polynomial with integer coefficient

I'm struggling with this question: Suppose $P(x)$ is a polynomial with integer coefficients such that non of the values $P(1),...,P(2010)$ is divisible by $2010$. Prove that $P(n)\neq 0$ for all ...
2
votes
3answers
91 views

Divisor in $\mathbb{C}[X]$ $\implies$ divisor in $\mathbb{R}[X]$?

let $P \in \mathbb{R}[X]$ be a real polynomial divisible by a polynomial $Q \in \mathbb{R}[X]$ in $\mathbb{C}[X]$. How can I easily show that $P$ is also divisible by $Q$ in $\mathbb{R}[X]$? A simple ...
0
votes
3answers
42 views

Euclidean algorithm in the ring of polynomials over a field

I need some help with the following division proofs. I suppose my biggest problem is not being able to visualize the algebra for one GCD equaling another GCD. I'm not sure of how to arrange the ...
3
votes
4answers
97 views

Show that $(x + 1)^{2n + 1} + x^{n + 2}$ can be divided by x^2 + x + 1 without remainder

I am in my pre-academic year. We recently studied the Remainder sentence (at least that's what I think it translates) which states that any polynomial can be written as P = Q*L + R I am unable to ...
1
vote
4answers
40 views

An exercise regarding polynomials

I guess it is a simple exercise though I'm not very good at polynomials. It asks: Find $m,n,p,q$ natural numbers such that the polynomial $X^m+X^n+X^p+X^q$ is divisible by $x^3+x^2+x+1$. Thank you in ...
2
votes
1answer
42 views

Show that $(x-a,x-b)=1$

Knowing that $K$ is a field, $a,b \in K$ different from each other,show that $x-a,x-b$ co-primes. We suppose that $\exists f(x) \in K(x)$ such that: $f(x)|x-a$ and $f(x)|x-b$ Then $\deg f(x) \leq ...
5
votes
3answers
159 views

Prove $x^3-3x+4$ is irreducible in $\mathbb{Q}[x]$

I want to prove $x^3-3x+4$ is irreducible in $\mathbb{Q}[x]$. Eisenstein's criterion doesn't apply here, so I think the simplest method would be to use the Rational Roots Test, right? If I can use ...
1
vote
1answer
38 views

GCD for multivariable polynomial ring

I'm reading Lectures on Modules and Rings by T. Y. Lam. It's on page 32 of the book, example 2.19A. It reads: (2.19A) Example. Let $k$ be a field. Then in the commutative polynomial ring $R = ...
2
votes
1answer
70 views

Relation between divisibility of polynomials in different rings, $h | f$ in $\mathbb{Z}[x], \mathbb{Z}/p^k\mathbb{Z}[x]$ and $\mathbb{F}_p[x]$

Let $p$ be a prime, $k$ a positive integer. Let $f,h \in \mathbb{Z}[x]$ be polynomials such that $h | f \mod p^k$ in $ (\mathbb{Z}/p^k\mathbb{Z})[x]$ $h \mod p$ is irreducible in $\mathbb{F}_p$ ...
1
vote
2answers
48 views

Finding a gcd in the form of a monic polynomial

The question is to find the greatest common divisor (in the form of monic polynomial) in $\Bbb F_5[X]$ of $f=x^2-x+4$ and $g=x^3+2x^2+3x+2$ I used the Euclidian Algorithm for polynomials and found ...
4
votes
3answers
86 views

Why doesn't this calculation work?

I want to find some closed form for $\gcd(x^3+1,3x^2 + 3x + 1)$ but get $7$ which is not always true.
0
votes
3answers
45 views

Using long division on polynomials

Can anyone show me how to find $x^5 + 1$ divided by $x^3 + 1$?
4
votes
2answers
39 views

Find all the polynomials $p \in \mathbb R [X]$ such that $(x+1)p=(p')^2$

(Where $p'(x)$ is the derivative of $p(x)$) Research effort: what I thought is that given that $(x+1)|(p')^2$ then $(x+1)|(p')$ (I'd like to justify better this, but I don't know how) Then, ...
1
vote
2answers
56 views

If I remove the premise $a\neq b$ in this question, will the statement still be true?

I encountered this proving problem, I can do the proof but my question is why in the condition/premise we need $a$ to be unequal to $b$? My guess is that even $a=b$, the statement is still true, is it ...
0
votes
2answers
61 views

Simple yet confusing: if $ f^2(x)=g^2(x)(x^2+1) $ then $gcd( f^2(x),g^2(x))=(x^2+1)$?

As mentioned in the title: f(x) and g(x) are polynomials above the Rationals field. if $ f^2(x)=g^2(x)(x^2+1) $ then does it mean that $ gcd( f^2(x),g^2(x))=(x^2+1) $? or maybe it isn't the ...
0
votes
3answers
30 views

Divisibility of polynomials in $\mathbb{Z}_n[x]$

For what values of $n$ is $x^2+1$ a factor of $x^5+5x+6$ in $\mathbb{Z}_n[x]$? I know how to divide in $\mathbb{Z}[x]$ (with long division), but what should I do here with $\mathbb{Z}_n[x]$, and it's ...
0
votes
1answer
82 views

Proof polynomial is always divisible by number

Given $f(x) \in \mathbb{Z} [x] $ a polinomyal, that evaluated in any $a \in \mathbb{N} $, results allways in a multiple of 101 or a multiple of 107 (both prime numbers). Prove then, that $f(x)$ it's ...
0
votes
2answers
127 views

Polynomial divisibility

Given $p(x) \in \mathbb Q[x] $ an irreducible polynomial, and $\alpha \in\mathbb C $ root of $p(x)$. Prove that if $q(x) \in \mathbb Q[x]$ it's a polynomial, such $q(\alpha) = 0$ then $p(x) \mid ...
0
votes
0answers
33 views

Question about zero-divisors , rings and polynomials.

Let $i,n,m$ be positive integers. For every nonnegative integer $k<i+1$ , let $a_k$ be elements of a ring $A$ that satisfies : $1)$ The ring $A$ is isomorphic to ${\Bbb R}^{n}\times{\Bbb ...
3
votes
2answers
37 views

A question on greatest common divisor

I had this question in the Maths Olympiad today. I couldn't solve the part of the greatest common divisor. Please help me understand how to solve it. The question was this: Let $P(x)=x^3+ax^2+b$ and ...
1
vote
2answers
65 views

Show that the matrix is invertible

let $A \in M_n(F)$ be a n by n matrix with values from an unknown field $F$. $P_A(t)$ is the characteristic polynomial of $A$, and $g(t) \in F[t]$ a polynomial of an unknown degree. assume that ...
0
votes
3answers
283 views

Prove $x^n-1$ is divisible by $x-1$ by induction

Prove that for all natural number $x$ and $n$, $x^n - 1$ is divisible by $x-1$. So here's my thoughts: it is true for $n=1$, then I want to prove that it is also true for $n-1$ then I use long ...
1
vote
1answer
39 views

polynomial division, gcd, question

We are asked to show that there are polynomials $p,q \in Q[t]$ such that: $p(t)*(t^4+2t^2+1)+q(t)*(t^4-3t^2-4) = t^2+1$ Is the answer the same for $t+5$ instead of $t^2+1$? What I tried doing: I ...
3
votes
1answer
83 views

Prove that $x^4+x^3+x^2+x+1 \mid x^{4n}+x^{3n}+x^{2n}+x^n+1$

Problem: Prove that $x^4+x^3+x^2+x+1$ divides $x^{4n}+x^{3n}+x^{2n}+x^n+1$ for all positive $n$ that are not multiples of $5$. I'd like to get some pointers about how to solve this. No full ...
2
votes
1answer
245 views
4
votes
1answer
155 views

Solve $a^3 + b^3 + c^3 = 6abc$

Find solutions for $a^3 + b^3 + c^3 = 6abc$ in $\mathbb{N}$, such that $gcd(a,b,c) = 1$, except for $(1,2,3)$ and its permutations. Using trial and error I found out that if $a,b,c$ are solution ...
3
votes
1answer
62 views

Solving $\frac{{2{x^3} - 11x + 6}}{{x - 2}}$ using algebraic juggling

Answer: $\eqalign{ & \frac{{2{x^3} - 11x + 6}}{{x - 2}} = \frac{{2{x^2}(x - 2) + 4{x^2} - 11x + 6}}{{(x - 2)}} \cr & = 2{x^2} + \frac{{4x(x - 2) - 8x + 11x + 6}}{{x - 2}} \cr ...
3
votes
1answer
118 views

Relationship between divisibility of polynomials and divisibility of its evaluations

Let $f$ and $g$ be primitive polynomials over $\mathbb{Z}$. Decide if the following is true: $f(x) \mid g(x)$ for infinitely many $x\in\mathbb Z$ implies $f\mid g$ as polynomials in ...
1
vote
2answers
391 views

Why should we append zeros during CRC calculation?

Say we have M as message bits , why do we need to append r-zeros to M message bits before performing the division to obtain r-bit checksum. Why don't we directly perform the division on the M message ...
1
vote
1answer
80 views

Divisibility of Multivariate Polynomials with Common Roots

Let $f(x_1,...,x_n)$ and $g(x_1,...,x_n)$ be polynomials in $\mathbb{C}[x_1,...,x_n]$ such that all roots of $f$ are roots of $g$ as well (but not necessarily viceversa). The question is: Does $f$ ...
2
votes
1answer
131 views

Determine the degree of a polynomial in a remainder theorem identity

How does one determine the degree of a polynomial in a remainder theorem identity without using long division? For example, a question asks: Divide $2x^2 + 4x +5$ by $x^2-1$ Writing the remainder ...
1
vote
3answers
303 views

Dividing polynomial by binomial using remainder theorem

An A level text book claims that one can find the quotient by first: 1.) Setting up an identity, $f(x)≡ Q(x)(divisor) + remainder$ 2.) Finding the coefficients However, another A level text book ...
2
votes
1answer
69 views

How can I get polynomial $p(x)$?

$p(x)$ is divided evenly into $x^{2}+1$, and $p(x)+1$ is divided evenly into $x^{3}+x^{2}+1$. How can I get $p(x)$?
0
votes
1answer
58 views

How much can we “reduce” this polynomial division?

Let's say we start with a univariate polynomial, $p(x)$: $$p(x) = x^n-1$$ We can then divide by another polynomial; for instance $q(x)$. For example, if $p(x) = x^6-1$ and $q(x) = x^2-x+1$ we have: ...
2
votes
1answer
139 views

Are there any limitations to the remainder theorem?

Does the remainder theorem only work for polynomial equations being divided by a binomial of the form $\ x-a\ $? Are there any limitations on the remainder theorem? I realize in polynomial division, ...
3
votes
3answers
138 views

Prove that $f(x)=0$ has no rational solutions

$f(x)$ $\in$ $Z[X]$ monic polynomial of degree $n$ $k,p$ $\in$ $N$ If none of the numbers $f(k), f(k+1), \ldots , f(k+p)$ is disivible by $p+1$, then $f(x)=0$ has no rational solutions.
1
vote
4answers
83 views

Find the greatest common divisor of the polynomials:

a) $X^m-1$ and $X^n-1$ $\in$ $Q[X]$ b) $X^m+a^m$ and $X^n+a^n$ $\in$ $Q[X]$ where $a$ $\in$ $Q$, $m,n$ $\in$ $N^*$ I will appreciate any explanations! THanks
0
votes
2answers
175 views

Division of a cubic equation by one of its factors [duplicate]

I'm trying to divide a cubic equation by a factor. This is the equation: $$ -\lambda^3 -\lambda^2 + 10 \lambda - 8 = 0$$ and this is the factor : $(\lambda - 1)$ I Googled about it and I found the ...
1
vote
1answer
1k views

How to calculate Inverse Z-Transform by long division

I am studying Feedback Control of Computing Systems. (specifically using Hellerstein's book, section 3.1.4, page 74) An inverse Z-Tranform also can be obtained by a long division. In the book there ...
2
votes
1answer
482 views

Calculating CRC code

I think I may be under a misconception. When calculating the CRC code, how many bits do you append to the original message? Is it the degree of the generator polynomial (e.g. x^3+1 you append three ...