Polynomials are expressions like $15x^3 - 14x^2 + 8$. Questions tagged with this concern common operations on polynomials, like adding, multiplying, polynomial long division, factoring and solving for roots.

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Sum of ideals in polynomial rings

Let $ I = \lbrace g(X) \in \mathbb{Z} \;| \; g(0)\in 5\mathbb{Z}\rbrace$ Show that $I$ is an ideal in $\mathbb{Z}[X]$, and that $I = \langle 5\rangle + \langle X\rangle $. From previous parts of ...
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2answers
59 views

State True or false? Does it have at least one zero in the interval?

The statement is as follows: If $\frac{a_{0}}{5}+\frac{a_{1}}{4}+\frac{a_{2}}{3}+\frac{a_{3}}{2}+a_{4}=0$ with $a_{4}\neq0$, then the equation $a_{0}x^{4}+a_{1}x^{3}+a_{2}x^{2}+a_{3}x+a_{4}=0$ ...
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2answers
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Polynomial ideals

I got stuck with an exercise while preparing for my exam, and could use a hint or two to move on... Let $f(X) = a_n X^n+a_{n-1} X^{n-1}+ \cdots +a_0 \in \mathbb{Z}[X]$ with $a_0\neq 0$ Assuming that ...
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1answer
169 views

What is the congruence class of $x^3\mod x^3+x+1$?

I have a given Polynom congruence with a Polynom $x^3+x+1$ ... so the set of the congruence classes is $\{0, 1,x,x+1,x^2,x^2+1,x^2+x,x^2+x+1\}$ But what would look this like? $$x^3\mod x^3+x+1\equiv ...
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Need hints on the following algebra problems.

I've been looking at these for over an hour and I don't understand how to do them. Any hints would be greatly appreciated. Let $p(x) = x^3 + x + 1$ and $F = Z_3[x]/\langle p(x)\rangle$. Factor ...
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1answer
16 views

Polynomial divisibility of a minimum polynomial

I am considering $x^2-x-1$ over $F=\mathbb{Z}/2=\{0,1\}$ as the minimum polynomial of a $3\times 3$ matrix $A$ with entries from $F$. I know that the characteristic polynomial (in this case a degree ...
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1answer
822 views

Relative Maxima/Minima of polynomial functions

I am taking the Pre Calculus 12 course online. I came across this concept that the online material teaches in 3 different ways, and each one contradicts the other. I find this extremely frustrating. ...
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1answer
29 views

relation of degree to number of zeros and Riemann hypothesis. [on hold]

Assume I have a single variable polynomial with degree n, is there a proof that there are n roots? If there is why hasn't this been applied to the Riemann hypothesis?
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5answers
949 views

How to rewrite $7-\sqrt 5$ in root form without a minus sign?

How to rewrite $7-\sqrt 5$ in root form without a minus sign ? For clarity "root form " means an expression that only contains a finite amount of positive integers , additions , substractions , ...
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Factorising $x^7-7x^6+21x^5-35x^4+35x^3-21x^2+20x+14$ into $\Bbb Q$-irreducible factors (Eisenstein's Criterion)

Factorize $x^7-7x^6+21x^5-35x^4+35x^3-21x^2+20x+14$ into $\Bbb Q$-irreducible factors. I've made the substitution $y=x-1$. So I get $y^7+13y+28$ which satisfies Eisenstein's Criterion for $p=13$. ...
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1answer
71 views

How can I find the solutions of this third degree equation?

I can't find the solutions of the following third degree equation: $$4 \lambda^3 + 4 \lambda^2 - \lambda -1 =0$$ with Ruffini's rule. Can someone help me find $\lambda_{1,2,3}$? Thank you for any ...
6
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1answer
133 views

Show that $x^n + x + 3$ is irreducible for all $n \geq 2.$

So first of all, I used the following from my lecture notes: If $f \in \mathbb{Z}[x]$ is primitive (gcd of all the coefficients is 1) - $f$ irreducible in $\mathbb{Z}[x] \Leftrightarrow$ f ...
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2answers
137 views

Some challenging Series, maximum value and polynomial factor questions

So I realize that the questions I am gonna ask are going to be a minute's work for some of you but I couldn't do them even after hours of searching for methods or something. They are from a ...
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2answers
110 views

Can anyone solve this equation?

Having trouble working this one out: $$25^x + (2 .5)^x = 35$$ Any help would be appreciated.
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1answer
16 views

Irreducibility of polynomials of a certain kind

Let us look at factorization over the integers of polynomials of the form $x^n+n$. For the first few values of $n$ we get $n+1$ - irreducible $n^2+2$ - irreducible $n^3+3$ - irreducible $n^4+4$ - ...
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1answer
18 views

The linear factor of the polynomial

Recently I've started to study polynomials, when I found out about the remainder and factor theorems as a way to avoid long polynomial division I couldn't understand the reason for every linear factor ...
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1answer
40 views

Algebraic tricks like componendo dividendo

I wanted to know all the operations that can be performed if say- a/b=c/d And Also on a/b=c/d=e/f Where a b c d e and f are various variables. One is if we add and subtract 1 we get (a+b) ...
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3answers
80 views

Show that $P(X) -X$ divides $P(P(X))-X$

Let $P$ be a polynomial in $R[X]$. Then show that $P(X) -X$ divides $P(P(X))-X$
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1answer
23 views

Algebraically Independence of polynomials.

Are the polynomials $(xy+xt+zt)t$, $(x+z)t^2$, $(x+z)(y+t)t$, $(y+t)(xy+xt+zt)$ algebraically independent ? If not what are all the relations between them. I tried to compute the determinant of the ...
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2answers
59 views

How to find an alternate form of this polynomial (factorize?)

I am trying to find the limit of the function $$\lim_{t \to 1} {{t^3-2t+1}\over{t^3+t^2-2}}$$ And it obviously evaluates to ${0\over0}$ so at first glance it is indetermined. But I have these two ...
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3answers
59 views

Ring isomorphism (polynomials in one variable)

Find an explicit isomorphism of the map $h: \Bbb F_5[x]/(x^2+x+2) \to \Bbb F_5[x]/(x^2+4x+2)$ I take the question to mean showing the 2 rings are isomorphic by finding an explicit mapping of each ...
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2answers
125 views

How to calculate $(1+x)(1+x+x^2)\cdots(1+x+x^2+\cdots+x^n)$

I have a combinatorics problem and I've reduced it to finding coefficient that stands with $x^n$ in this polynomial, $$(1+x)(1+x+x^2)...(1+x+x^2+...+x^n)$$ But now I'm stuck. Can someone help me ...
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0answers
25 views

Prove or disprove an inequality related to zeros of polynomials [closed]

If a polynomial $p(z)$ of degree $n$ with zeros $z_1,z_2,\cdots,z_n$ assumes maximum at $w$ on $|z|=1.$ Prove or disprove that the Harmonic mean of $|z_k-w|,$ $k=1,2,\cdots,n$ is greater or equal to ...
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0answers
37 views

why is $ F [x]/(x^2 -1)$ isomorphic to $ F \times F $ [duplicate]

why is $ F [x]/(x^2 -1)$ isomorphic to $ F\times F $ I know F is a field where 1+1 can't equal 0 I've calculated the idempotents as $1/2 x+1/2$ and $-1/2 x+1/2$ not sure what to do next
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1answer
17 views

Show that there is $z_0$ on the unit circle such that $|p(z_0)|>1$.

Let $p(z)=z^n+a_{n-1}z^{n-1}+\dots+a_0$. Prove that if $p(z)\neq z^n$, then there is $z_0$ on the unit circle such that $|p(z_0)|>1$. Hint: Consider $q(z)=z^np(1/z)$. I have really been messing ...
2
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2answers
576 views

Approximate a polynomial function using a sum of sine waves

I have a polynomial function which I need to approximate by a sum of sine waves with constant amplitude along a given domain. From what I hear, this might be a good time to make use of Fourier ...
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0answers
17 views

Proof of inequality of polynomials

Assume F, G, M, and N are Laurent polynomials over $\mathbb{R}$, with coefficients either 0 or 1. Assume that $F \neq M$, and that we have $$F(X^{2^n})G(X^{-1})=M(X^{2^n})N(X^{-1}),$$ for $n \in ...
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0answers
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Bound on partial derivatives when the total derivative is bounded.

Let $f\in\mathbb{Z}[x,y]$ and fix $k\in\mathbb{N}$. Suppose that for all $v=(v_1,v_2)\in S^1$, $$ \left|\frac{d^k}{dt^k}f(tv)\right|_{t=0} $$ is bounded by $C$. (The value of $C$ that I use comes ...
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0answers
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Finding ring isomorphism [duplicate]

Find isomorphism from $F_5[x]/(x^2+x+2) \rightarrow F_5[x]/(x^2+4x+2) $ I realise both polynomials are irreducible therefore form fields, not sure how to form a isomorphism from one to the other, help ...
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Quadratic Polynomial factorization

This could be primary school stuff. But I want to ask it. In factoring $x^2+bx+c$ (i.e. $a = 1$ in $ax^2+bx+c$), we find $m$ and $n$ such that $m+n = b$ and $mn=c$. We can reason this well as ...
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1answer
71 views

Find the sum of all real numbers x such that [closed]

Find the sum of all real numbers $x$ such that $$ 5x^4-10x^3+10x^2-5x-11=0. $$
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1answer
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Polynomial long division modulo 7,

I need to determine the quotient and remainder using polynomial long division in $Z_7[x]$. I'm not sure how to tackle it with the polynomials given, and I'm growing frustrated by it. I need to divide ...
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3answers
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A divisibility question concerning polynomials.

What is the condition that $$x^2 + x + 1$$ is a factor of $$(x + 1)^n − x^n − 1$$.
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The meaning of the term “lies” above the x-axis

On my math quiz there was an equation that produced a polynomial that starts positive, crosses the x axis at negative 3, and then stays negative. There was a true or false question that asked if the ...
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1answer
25 views

Advice to solve a system of 8th order univariate polynomials

I am struggling to solve a least square problem in which the tedious part is the initialization. Grid search methods are out of question. The initial problem I've stated my problem in a previous ...
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2answers
53 views

Given 2 points where a line and a curve cross, find the third point where line and the curve cross.

The curve $y^2 = x^3 + 8$ contains the points $(1,-3)$ and $($$-7\over4$,$13\over8$$)$. The line through these two points intersects the curve in exactly one other point. Find this third point. P.S. ...
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Roller Coaster, Where does the ride start? [closed]

Question: 2. The Giant coaster is modeled by h(t) = t^3 - 15t^2 + 44t The height of the coaster can be determined for the first 12 seconds of the ride by this polynomial. Calculate maximum and ...
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4answers
393 views

The existence of a polynomial of degree $n$ with $m$ real roots when $m\equiv n \pmod 2$.

I came across the following fact: Let $m$ and $n$ be natural numbers such that $n\geqslant m$ and $m\equiv n \pmod 2$. Then, there exists an irreducible polynomial $f\in\mathbb{Q}\left[X\right]$ ...
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0answers
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How to get the sum of the bernstein approximation polynomials coefficients

As $(u_{j},v_{j},w_{j})\in\triangle_{2}$, $\triangle_{2}$ is two dimensional simplex, for $u_{j}^{\lambda_{j_{1}}}v_{j}^{\lambda_{j_{2}}}w_{j}^{\lambda_{j_{3}}}$ defined on the $\triangle_{2}$, the ...
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1answer
21 views

Sequence of elements $x_i \in \bar{Q}$

I am reading about applications of Galois theory to polynomials, but when using it for a degree 5 polynomial I got confused by the following: I need to show that any sequence of elements $x_i \in ...
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1answer
307 views

Krylov-like method for solving systems of polynomials?

To iteratively solve large linear systems, many current state-of-the-art methods work by finding approximate solutions in successively larger (Krylov) subspaces. Are there similar iterative methods ...
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1answer
25 views

Find a sequence of complex polynomials with certain properties. (Hardy spaces over unit circle)

Let $\lambda\in \Bbb S^1$. Find a sequence of complex polynomials $p_n(z)$ such that for any $c>0$ the following inequality does not hold: $$|p_n(\lambda)|\le c\cdot \|p_n\|$$ where ...
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1answer
28 views

Stone Weierstrass theorem generalization

Find all such functions $g:[a,b]\to [a,b]$ such that $g$ is continuous. For any continuous function $f:[a,b]\to \mathbb{R}$, given $\varepsilon >0$ there is a polynomial $P_{\varepsilon}(t)$ ...
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4answers
250 views

Doubt on 'Rest Theorem' in Polynomial Division

I'm dealing with an issue dividing polynomials, I have: Determine the value of $a$ to make: $x^2 + 2x - a$ divisible by $x + 4$ I don't know even where to start, this $a$ confuses me a lot; Thanks ...
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2answers
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Existence of complex polynomial with modulus on $|z|=1$ less than 1

I wonder if there exists a complex polynomial $P(z),z\in \mathbb{C}$ s.t $$\forall |z|\leq 1, P(z)<1.$$ I know that using modulus maximum principle, we only need to find $$P(z)<1, \forall ...
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0answers
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Magnitudes of roots of random polynomials

I'm looking at the roots of random polynomials with integer coefficients, and constant term=leading term = 1. Using the Mathematica code ...
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1answer
20 views

Approximate a positive multivariate function with a sum of squares of polynomials?

I am constructing approximation to a multivariate function which I know is positive. My question is the following: Let $f(x)$ be a multivariate positive and continuous function. Can we approximate ...
71
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1answer
1k views

Is There An Injective Cubic Polynomial $\mathbb Z^2 \rightarrow \mathbb Z$?

Earlier, I was curious about whether a polynomial mapping $\mathbb Z^2\rightarrow\mathbb Z$ could be injective, and if so, what the minimum degree of such a polynomial could be. I've managed to ...
2
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1answer
173 views

By viewing the polynomials as a difference of two squares, factorise the following polynomials.

By viewing the polynomials as a difference of two squares, factorise the following polynomial: $$x^4+x^2+1.$$ I searched but couldn't find a way to solve this Edit: By using Hans Lundmark hint, I ...
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2answers
28 views

How can I prove this operator is not continuos

Let $X$ the normed space of all polynomials on $J=[0,1]$ such that $||x||$=max$|x(t)|$ $t \in[0,1]$ and we have the following operator $Tx(t)=x'(t)$ prove this operator is not continuous