This tag is used for both basic and advanced questions on polynomials in any number of variables. Including, but not limited to: solving for roots, factoring, checking for irreducibility. This tag is rarely used as the only tag for a question.

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1answer
33 views

Divided differences of a polynomial

Let $f(x)$ be a polynomial of degree $ n $. Prove that the $k$-th divided difference $f[x_0, x_1,... x_k]$ is a polynomial of degree $n-k$, with respect to one of the variables $x_0, x_1, ... x_k$. ...
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0answers
31 views

Size of coefficients of polynomials that satisfy a Chebyshev-like extremal property

The famous Chebyshev polynomials satisfy many extremal properties. One of these is that they attain the largest possible derivative over the interval [-1,1] among polynomials whose absolute value over ...
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2answers
25 views

Finding roots of cubic (trig)

The question is By putting $x$ $=$ $\frac 23 cos (\theta)$ Find the exact roots of the equation in terms of $\pi$ $$ 27x^3 - 9x = 1 $$ What I have attempted: $$ ...
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1answer
13 views

S-polynomial remainder of two polynomials

In Page 90 of Ideal, Varieties and Algorithms, it shows a calculation of two polynomials being $x^3-2xy$ and $-2xy$, and shows the remainder being zero when using the S-polynomial method, but whenever ...
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1answer
18 views

Comparing a series expansion to polynomial regression

So I don't have a great background in mathematics but I have a quick and hopefully simple question for you guys. I'm a graduate student and I'm doing some polynomial regression on some thermodynamic ...
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2answers
419 views

Symbolic polynomial interpolation

I'm trying to create polynomials from some symbolic points to discretize derivations. This means I'm having data like $(a, \phi(a)),\ (b, \phi(b) ) $and $(c, \phi(c))$ and want to fit a second order ...
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2answers
26 views

Is there a concrete definition/formula for finding the leading coefficient of any polynomial?

Is there a concrete definition that tells one the leading coefficient of any polynomial? Using logic, I derived this formula: $$ a=\frac{\frac{d^p}{dx^p}f(x)}{p!}$$ where $f(x)$ is a polynomial, $p$ ...
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1answer
25 views

Why is the set $U$ with $p(0)=c$ only a subspace of $P_3$ when $c\in \mathbb{R}=0$ ?

I'm having trouble grasping why the set $$ U_c = \{ p \in P_3\ |\ p(0) = c \}$$ with $c\in \mathbb{R}$ only counts as a subspace of $P_3$ when $c=0$. I've been told that it wouldn't be a valid ...
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2answers
90 views

Is $‎‎‎\sqrt[3]{y^3}‎‎‎$ or $\frac{x^2}{x}$ a polynomial?

A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. Now are $$‎‎‎\sqrt[3]{y^3}‎‎‎,\quad ...
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0answers
41 views

Perturbing Coefficients of Polynomials

I'm stuck on 7. (c) (see problem below) proving that this polynomial has no more than $m$ roots (so far I have that it has $\geq m$ roots). I am trying to proceed by contradiction, assuming that there ...
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0answers
32 views

The sum of all primitive $n$-th roots of unity in the algebraic closure of $F(x)$

If $n$ is any natural number, then $\mu(n)$ is the sum of all primitive $n$-th roots of unity in $\mathbb{C}$ where $\mu$ is the Möbius function defined as $\mu(n)=0$ if $n$ is not square free and ...
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1answer
39 views

Prove $x^n+1$ is irreducible over $\mathbb{Q}[X]$ iff $n=2^k$ for $k \in \mathbb{N}$

Unfortunately, I cannot find any information on or anything similar to this particular question. Might be quite new. In all honesty, I don't know how to tackle either side of this question. By ...
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3answers
48 views

State whether $x^5-5x^4+10x^3-7x^2+8x-4$ is irreducible or not

I have tried everything in my knowledge and no, I cannot state it. I have tried a factorizor online which tells me that it is not factorizable hence irreducible. But I cannot reason why. I looked at ...
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0answers
28 views

Is it always possible to find the roots of $P(z)=az^4+bz^3+cz^2+bz+a$, where $a,b,c \in \mathbb{R}^*$, by first dividing both sides by $z^2$?

A classic way to solve quartics in the form $P(z)=az^4+bz^3+cz^2+bz+a$, if we know that the roots lie on the unit circle, is to divide both sides by $z^2$ and then use the fact that if $$z=\cos \theta ...
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1answer
35 views

The value of nested square roots

Find the value of $$x=\sqrt{11-\sqrt{11+\sqrt{11-\sqrt{11+\cdots }}}}$$ we can write $x$ as $$x=\sqrt{11-\sqrt{11+x}}$$ Squaring both sides we get $$x^2=11-\sqrt{11+x}$$ or $$11-x^2=\sqrt{11+x}$$ ...
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1answer
42 views

Is every polynomial with integer coefficients prime in $\mathbb{Z}[x]$ also prime in $\mathbb{Q}[x]$? [closed]

The question is as in the title: Is every polynomial with integer coefficients prime in $\mathbb{Z}[x]$ also prime in $\mathbb{Q}[x]$?
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0answers
26 views

Polynomials : Indicate if a is a root non-existent for P(x) then$\frac 1a$is a root

We have $$ P(x)=2x^3+3x^2-3x^1-2 $$ 1- Calculate P(-2) & P(1) & P(3) (already answered) 2- Do the Euclidean division for P(x) on $$x +\frac 12$$ (already answered) 3- Indicate if a is a ...
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2answers
42 views

Find polynomial in equation

Find polynomial $f(x)$ with real coefficients that satisfied: $$x^2f(x) + 2 = f(x^2) + 2x^3$$ I find that $\deg f$ can be $1$ and $2$. $$\deg f = n $$ $$2+n=\max(2n,3)$$ First case $2=n$ Second ...
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1answer
489 views

Working with casus irreducibilis

I read about casus irreducibilis here. As an example of casus irreducibilis, it says we can factor $x^3 - 15x - 4$ to find $4$ as a root and it also has two other real roots. Using Cardano's method we ...
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1answer
60 views

An equation in $\mathbb Z[x]$

Let $$p(x)=a_nx^n+a_{n-1}x^{n-1}+……+a_2x^2+a_0;\space\space (a_1=0)$$ be a polynomial in $\mathbb Z[x]$; let $\mathcal S$ the set of all such polynomials. It is considered the equation ...
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2answers
70 views

Reliable test for intersection of two Bezier curves

Is there a test which reliably decides whether two Bezier curves intersect or not? I don't need to know how many intersections there are or at what parameters they appear at. I just would like to ...
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3answers
6k views

Root or zero…which to use when?

This may seem like a very basic question, but: What exactly is the difference between a root of a polynomial, and a zero? Of course I realise that they are technically exactly the same thing, but ...
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1answer
40 views

Why is $x^3 + 3x + 2$ irreducible by plugging in elements of $\mathbb{Z}_5$

The polynomial $x^3+3x+2$ is irreducible in $\mathbb{Z}_5[x]$. I get that it must take the form of $(x-a)g(x)$ where $a$ is a zero,but my book plugged in elements of $\mathbb{Z}_5$ to show no zeros, ...
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2answers
50 views

Show that $x^5-x^2+1$ is irreducible in $\mathbb{Q}[x]$.

Show that $x^5-x^2+1$ is irreducible in $\mathbb{Q}[x]$. I tried use the Eisenstein Criterion (with a change variable) but I have not succeeded. Thanks for your help.
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0answers
14 views

Negative polynomial remainder when calculating CRC

I ran this in Wolfram Alpha: (x^15+x^13+x^11+x^9+x^7+x^5+x^4)/(x^4+x^2+1) Got -x^2-1 as remainder. This is R(x), which corresponds to my FCS bits, right? But it is negative. x^2+1 is 101, fine, but ...
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1answer
28 views

Explain Multidegree of a polynomial

Definition (The book Ideals, Varieties and Algorithms, Cox et al, pages 59-60 on 2008 edition): Let $f=\sum_a a_a x^a$ be a nonzero polynomial in $k[x_1,\ldots, x_n]$ and let > be a monomial ...
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1answer
29 views

Let $P(x)$ be a polynolmial with degree $2009$ and leading coefficient unity such that $P(0)=2008,P(1)=2007,P(2)=2006,\ldots,P(2008)=0$,

Let $P(x)$ be a polynolmial with degree $2009$ and leading coefficient unity such that $P(0)=2008,P(1)=2007,P(2)=2006,\ldots,P(2008)=0$,then the value of $P(2009)=n!-a$ where $n!$ is $n$ ...
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1answer
31 views

Define $P(x)\in [\mathbb{R}]$ is a polynomial that satisfy $P(x).P(y)=P(\frac{x+y}{2})^{2}-P(\frac{x-y}{2})^{2}$

Define $P(x)\in [\mathbb{R}]$ is a polynomial that satisfy $P(x).P(y)=P(\frac{x+y}{2})^{2}-P(\frac{x-y}{2})^{2}$ (1) Some notes may help you. It is very clear that with y=0 then ...
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3answers
52 views

To find number of roots in interval $(1,2)$

To find number of roots in interval $(1,2)$ $$f(x)= 3x^2 - 12x + 11 + \frac{x^3-6x^2+11x-6}{5} $$ Now $f(1)f(2)<0$. So it has one root. But how do I know whether it has two or three or whatever. ...
3
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1answer
156 views

Polynomials and Arithmetic

Consider the polynomial $$p(x) = a_0 + a_1x + a_2x^2 + · · · + a_nx^n$$ where $a_0, a_1, . . . , a_n ∈ \Bbb Z$. Show that if $p(x_i) = 7$ for 4 distinct integers $x_0, x_1, x_2, x_3$, then $p(z) \neq ...
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1answer
94 views

Statistical problem: how many books of different widths fit it into a self of a limited certain width?

Let's say I have N sets of books, being the size of the books in a set the same. The cardinality of the every set is different: I might have 3 books of width 5 units (first set), 6 books of width 10 ...
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2answers
37 views

Define Derivative of Product of Polynomials

I have a a problem with defining a certain term... The derivative of a product of polynomials is the sum of derivatives of the products of the summands of the polynomials of the original product. ...
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1answer
27 views

Stuck with the statement: $t^4+2$ in $\mathbb{Z}_5$ gives rise to…

This is from Ian Stewart's book on Galois theory, I am looking at irreducible polynomials. It talks of irreducibility over mod. It takes as an example, $f(t)=t^4+15t^3+7$ over integers, and asks us ...
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1answer
7 views

If $E$ is the splitting field of a separable polynomial on $K$, then $E/K$ is normale.

Let $K$ a field and $E$ the splitting field of a separable polynomial $f\in K[X]$. Show that $E/F$ is normale. My definition of normale extension is that $E/K$ is normale if ...
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1answer
35 views

Finding all roots of multivariate polynomial using Newton's method

I read that it is possible to find a solution to a nonlinear system of equations using Newton method and Jacobian matrix. But if I understood correctly, this finds just one solution, and which one ...
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1answer
31 views

The dimension of the space of continuous functions that are piecewise polynomials of degree $k$

I am trying to calculate $$ \dim( \{ f\in C^0 ([a,b]) : f_{|[x_{j-1},x_j]} \in \mathcal{P}_k, j = 1,...,m \}) \text{ with }m,k \in \mathbb{N} $$ Is $m(k+1)$ correct? My thoughts: I have $m$ ...
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1answer
27 views

Division with remainder in $K[x,y]$

Let $K$ be a field and $f\in K[x,y]$ such that $a:=\frac{\partial f}{\partial x}(0,0)\not=0$. Let $b:=\frac{\partial f}{\partial y}(0,0)$. In a lecture of mine, it was claimed now that we can divide ...
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1answer
41 views

Factorizing $X^4-Y^2$ in $\mathbb{Q}[X,Y]$

I want to factorize $X^4-Y^2$ in $\mathbb{Q}[X,Y]$ in irreducible factors. I thought about using Eisenstein's criterium to show that it is irreducible, though I'm not sure what the prime element is ...
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0answers
60 views

A polytime language with no subsets of lesser time complexity

For any integer $l>0$ does there always exist a language with time complexity of order $O(n^l)$ such that it has no subsets of a lesser time complexity ie $O(n^m)$ for any $m< l$. We talk of ...
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2answers
41 views

How to find the polynomial given its factors? (A bit typical one here)

I recently saw this problem (and I should have paid more attention to my middle school maths classes). Find a 3 degree polynomial of $x$ which is $0$ when $x=1$ and $x=-2$, $4$ on $x = -1$ and $28$ ...
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3answers
50 views

Cubic equation (polynomial)

A cubic polynomial with real coefficients, $a x^3 + b x^2 + c x + d$, has either three real roots, or one real root and a pair of complex conjugate ones. If the latter happens, what is the explicit ...
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0answers
64 views

Elementary dvisibilty problem involving power sums.

Prove that $1^n + 2^n + \cdots + m^n$ does not divide $(1+2+ \cdots +m)^n$ for any even integer $n\geq 2$. For $n\leq 4$, the result easily follows from the relevant identities. For $n\geq 6$, i ...
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1answer
24 views

How do I determine the graph of functions involving radicals?

What is the explanation behind: the graph of $h(x)=\sqrt{4-x^2}$ is the upper half of the graph of $x^2+y^2=4$ the graph of $g(x)=-\sqrt{2-x}$ is the lower branch of the parabola $x=2-y^2$ I kind ...
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2answers
13 views

Real polynomial involving vector product of complex conjugates

Let $v \in \mathbb{C}^n$ be a complex vector and $\bar{v}$ be its complex conjugate, i.e. $v_i = \bar{\bar{v}}_i$. Let $x = (x_1, \dotsc,x_n)^T$ be a vector of variables. What is a simple argument to ...
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3answers
75 views

solve $t^5+t+1=0$

I honestly don't have any idea at all on how to solve this. I am asked to find solutions under $\mathbb{R},\mathbb{Q},\mathbb{C}$ respectively but this seems impossible to solve without a computer. ...
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5answers
68 views

Remainder Theorem - $(x+1)^{2015}$

This one just caught me without a clue. Find the remainder when $(x+1)^{2015}$ is divided by $x$. Assuming I don't use Binomial expansion. What are other alternatives?
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1answer
183 views

Polynomials with some roots whose product is 1

Consider the complex coefficient polynomial equation \begin{eqnarray} ...
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1answer
35 views

Irreducible polynomials in $\mathbb{C}[X,Y]$

I have the polynomial $X^2+Y^2-1$ in $\mathbb{C}[X,Y]$. Is this irreducible? If not, how do I factorize it? Should I handle this the same as if it were $\mathbb{R}[X,Y]$, or should I do it ...
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1answer
36 views

Isomorphic quotients of polynomial rings over finite fields

What are the elements of $\mathbb{F}_3[X]/(X^3-3)$? A similar question was posted here: Elements of the field $F_2[x] / (x^3 + x + 1)$, but it doesn't explain why the elements of that field look ...