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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1answer
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Roots of cubic equation

If$\frac{1+\alpha}{1-\alpha},\frac{1+\beta}{1-\beta},\frac{1+\gamma}{1-\gamma}$ are the roots of the cubic equation $f(x)=0$ where $\alpha,\beta,\gamma$ are the real roots of the cubic equation ...
3
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1answer
34 views

Understanding a simple proof about minimal polynomials

Let $T \colon V\to V $ be a linear operator, where $V$ is a vector space over $F$. Suppose that the minimal polynomial $M(t)$ of $T$ can be factored into the product of two coprime and monic ...
2
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1answer
50 views

Polynomial root finding: Bernstein vs Budan

Budan's and Vincent's theorems can be used to isolate the real roots of a real polynomial. I have read papers which compared it favorably to other root finding methods. However, roots can also be ...
3
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2answers
6k views

The degree of a polynomial which also has negative exponents.

In theory, we define the degree of a polynomial as the highest exponent it holds. However when there are negative and positive exponents are present in the function, I want to know the basis that we ...
42
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6answers
4k views

Continuity of the roots of a polynomial in terms of its coefficients

It's commonly stated that the roots of a polynomial are a continuous function of the coefficients. How is this statement formalized? I would assume it's by restricting to polynomials of a fixed ...
2
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0answers
40 views

Proving that $t^{p^r}-a$ is irreducible when $a\in k$ is not a $p$th power

Let $p$ be an odd prime, $F$ a field of characteristic $0$ and $a\in F$ with $a\neq 0$. Assume $a$ is not a $p$th power in $F$. Prove that for every positive integer $r$, $t^{p^r}-a$ is irreducible ...
0
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1answer
48 views

False positives with Descartes rule of signs

Descartes rule of sign can be used to isolate the intervals containing the real roots of a real polynomial. The rule bounds the number of roots from above, that is, it is exact only for intervals ...
4
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1answer
40 views

Polynomial GCD in the presence of floating-point errors

The crucial requirement for using root isolation methods based on Vincent's theorem is that the input polynomial does not have multiple zeros. One way to remove the multiple zeros is to use polynomial ...
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2answers
1k views

Solving 4th degree(quartic) polynomial with all irrational roots

I'm a self-learner trying to solve polynomial equations. This equation $$x^4-6x^3+\frac{31} 4x^2-\frac 3 2x-2=0$$ has $4$ irrational roots (if I'm right.) I found the solution by on-line solver. ...
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0answers
30 views

To construct a power series such that the radius of convergence of the power series $\sum_{n=0}^{\infty} a_n b_n x^n$ is $2R$.

Let $\sum_{n=0}^{\infty} a_n x^n$ is a power series with radius of convergence $R(>0)$. To construct a power series $\sum_{n=0}^{\infty} b_n x^n$, other than $\sum_{n=0}^{\infty} (\frac x2)^n$, ...
2
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2answers
99 views

Finding $4$ variables using $3$.

if I have: $ x=\dfrac{a-.5b-.5c+.25d}{a+b+c+d}$ $ y=\dfrac{\dfrac{b\sqrt{3}}{2}+\dfrac{c\sqrt{3}}{2}+\dfrac{d\sqrt{3}}{4}}{a+b+c+d}$ $ z=a+b+c+2d $ Then how do I get back to: $ a= $ , $ b= $ , $ ...
3
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1answer
44 views

Proof of Cohn's Irreducibility Criterion

I was looking for an elementary (or involving introductory level abstract algebra/analysis) proof of Cohn's Irreduciblity Criterion: If $$ a_0, a_1, \dots, a_n \in \Bbb{Z} $$ and $$ 0 \le ...
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4answers
95 views

coefficient of $x^{17}$ in the expansion of $(1+x^5+x^7)^{20}$

I found this questions from past year maths competition in my country, I've tried any possible way to find it, but it is just way too hard. find the coefficient of $x^{17}$ in the expansion of ...
1
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1answer
26 views

Prove that $T_n(x)={}_2F_1\left(-n,n;\tfrac 1 2; \tfrac{1}{2}(1-x)\right) $

Prove that, for Chebyshev polynomials of the first kind, \begin{align} T_n(x) & = \tfrac{n}{2} \sum_{k=0}^{\left \lfloor \frac{n}{2} \right \rfloor}(-1)^k \frac{(n-k-1)!}{k!(n-2k)!}~(2x)^{n-2k} ...
12
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2answers
229 views

Solve $x^7-5x^4-x^3+4x+1=0$ for $x$

Solve for $x$ $$x^7-5x^4-x^3+4x+1=0$$ This equation has been bugging me since the past few days. I have found, using the Rational Root Theorem that $x=1$ is a root of this equation. However, ...
4
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5answers
341 views

polynomial of fifth degree

Prove that the largest number of real roots of the equation $ x^5+a_1x^4+a_2x^3+a_3x^2+a_4x+a_5=0$ whose coefficients are real,is three if $2a_1^2-5a_2<0.$ My attempt is: As coefficients are ...
4
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3answers
298 views

Finding roots of cubic equation

If $\alpha,\beta,\gamma $ are the roots of the equation $2x^3-3x^2-12x+1=0$.Then find the value of [$\alpha$]+[$\beta$]+[$\gamma$],where [.] denotes greatest integer function. My attempt: I first ...
6
votes
3answers
394 views

Polynomial shift

Suppose there is a polynomial: $p(x) = a_0x^n + a_1x^{n-1} + ... + a_{n-1}x + a_{n}$ I would like to "shift" it (I'm not sure what is the right term), by substituting $x$ for some other function of ...
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1answer
34 views

Common solutions of two inequations

Find the real values of $a$ for which the inequations $x^2-4x-6a\leq 0$ and $x^2+2x+a\leq0$ have only one real solution common. My attempt: Let $\alpha$ be one real common root of two inequations. ...
10
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4answers
706 views

How to show that a polynomial does not have real roots

How to show generally that a polynomial does not have real roots. Well, for eg lets take the polynomial $x^8-x^7+x^2-x+15$ . Here the power($n=8$) is even so it can have real roots or it might not ...
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1answer
68 views

Prove that in the ring $F[t,t^{-1}]$ we have $x=t^n \Leftrightarrow x \mid 1$ and $t-1 \mid x-1$

I want to prove the following lemma: For any $x$ in the ring $F[t,t^{-1}]$ ($F[t,t^{-1}]$: the polynomials in $t$ and $t^{-1}$ with coefficients in the field $F$), $x$ is a power of $t$ if and ...
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0answers
33 views

$x^p -x-1$ irreducible over $\mathbb{F}_{p}$ [duplicate]

Show that $x^p - x -1$ is irreducible over $\mathbb{F}_{p}$. I've seen this polynomial (or some variation x^p -x -a) on several of our qualifying exams and in every case they ask you to show it is ...
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0answers
30 views

Finding roots of $4$th degree conjugate reciprocal polynomial

I am developing a computer program and the following polynomial, of which I need to obtain the roots, turned up $$Ax^4 + Bx^3 + Cx^2 + \overline{B}x + \overline{A}, \quad \text{where } A, B,x \in ...
3
votes
4answers
102 views

Solve $10x^4-7x^2(x^2+x+1)+(x^2+x+1)^2=0$

How to solve this equition? $$10x^4-7x^2(x^2+x+1)+(x^2+x+1)^2=0$$ My attempt: $$ 10x^4 - (7x^2+1)(x^2+x+1)=0$$ Thats all i can Update Tried to open brakets and simplify: $$(7x^2+1)(x^2+x+1) = ...
4
votes
2answers
233 views

Idempotents in a Quotient Ring

Let $R=\mathbb{Z}_p[x]/(x^p-x)$. Show that $R$ has exactly $2^p$ elements satisfying $r^2=r$. I know that for $f,g\in\mathbb{Z}_p[x]$, we have $f-g\in(x^p-x)$ if and only if $f(a)=g(a)$ for all ...
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0answers
33 views

What is ${\sigma _{\varepsilon ,W}}(P)$? [on hold]

Definitions: ${A_j},{\Delta _j} \in {C^{n \times n}},(j = 0,1,2....m)$ ${\rm{P(}}\lambda {\rm{) = }}{{\rm{A}}_m}{\lambda ^m} + .....{A_1}\lambda + {A_0}$ is a matrix polynomial, and $\lambda $ is ...
3
votes
2answers
127 views

Polynomial of 11th degree

Let $f(x)$ be a polynomial of degree 11 such that $f(x)=\frac{1}{x+1}$,for $x=0,1,2,3.......,11$.Then what is the value of $f(12)?$ My attempt at this is: Let ...
0
votes
2answers
36 views

How to reduce the multiplicity of existing real roots without introducing new real roots?

Given a monic polyomial $P(x)=x^d+r_{d-1}x^{d-1}+\cdots+a_1r+a_0\in\mathbb{R}[x]$ is there a way to manipulate the coefficients of $P$ in an algebraic way such that the new polynomial has exactly as ...
2
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1answer
33 views

Conformal mapping and its application in finding roots of polynomial

So for a polynomial, if we want to find the roots in a complex plane. Rouche's theorem is the first tool in my head. However, I saw several problems of finding the roots in the first quadrant or upper ...
39
votes
2answers
2k views

Decomposing polynomials with integer coefficients

Can every quadratic with integer coefficients be written as a sum of two polynomials with integer roots? (Any constant $k \in \mathbb{Z}$, including $0$, is also allowed as a term for simplicity's ...
7
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0answers
113 views

why this polynomials is non-negative? [closed]

show that this polynomials is non-negative $$f(x,y)=x^2(x^2-1)^2+y^2(y^2-1)^2-(x^2-1)(y^2-1)(x^2+y^2-1)\ge 0,\forall x,y\in R$$
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0answers
22 views

Are the polynomials that are orthogonal in the continuous case, still continuous in the discrete case?

One of my friends asked me this question. "Are the polynomials that are orthogonal in the continuous case, still continuous in the discrete case?" It is curious how even the most trivial questions ...
12
votes
1answer
333 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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votes
1answer
52 views

Find a polynomial for which a certain equality is true

For which real polynomials $p(x)$ does $p(p(x))+p(x) = x^4+3x^2+3$ for all real x.
4
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1answer
170 views

Question about quartic equation having all 4 real roots

I would appreciate if somebody could help me with the following problem.I am not good at quartic equations,so could not attempt much. Q:The number of integral values of $p$ for which the equation ...
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1answer
25 views

Polynomials and polynomial division [closed]

There exists a rational number z such that $(9x^3-10x^2-16x-4)/(x-z)$ can be written as a second degree polynomial $Ax^2+Bx+C$. Find z, A, B, C Can someone help me figure this out?
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1answer
28 views

Determine the order of $\exp Q(z)$ when $Q$ is a polynomial of degree $q$.

I am looking to determine the order of $f(z) = \exp Q(z)$ when $Q$ is a polynomial of degree $q$. I think the order is $q$, but I am struggling to prove it. The definition of order is: An entire ...
2
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1answer
104 views

How prove $\deg (f^{m} - g^{n})\ge \frac{mn - m - n}{n}\deg f + 1$?

Let $m,n\in \mathbb{N}, \ m,n\ge 2,$ Let $f,g$ be polynomials (with real coefficients, or more generally - complex) such that $f^{m} - g^{n}\neq 0.$ How prove $\deg (f^{m} - g^{n})\ge \frac{mn - m - ...
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2answers
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Polynomial problem solving: find the values of $a$ and $b$. [closed]

If $x^2 + 2x − 1 ≡ (x − 1)^2 + a(x + 1) + b$, find the values of $a$ and $b$. Answered: Option 1; given: x = 0, x = 1, find the expression or polynomial of b and then solve for a and b. Option 2; ...
2
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0answers
12 views

Question on real polynomial in projective space

Hi all I was given this question and desperately in need of help. I am given a homogeneous polynomial of degree 4 of two variables x and y, with real coefficients with 4 real distinct projective roots ...
3
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1answer
96 views

if $(1-a)(1-b)(1-c)(1-d) = \frac{9}{16}$ then minimum integer value of $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} = ?$

Given $a,b,c,d > 0$, how do we find the minimum integer value of $n=\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d}$ such that $(1-a)(1-b)(1-c)(1-d) = \frac{9}{16}$.
1
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1answer
75 views

Compactness of a set of matrix polynomials with a norm restriction

Suppose $P_\Delta (\lambda) = (A_m + \Delta _m)\lambda^m + \cdots + (A_1 + \Delta_1)\lambda^1 + (A_0 + \Delta_0)$ is a matrix polynomial, and $\lambda $ is a complex variable. $A_j,\Delta_j \in ...
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0answers
22 views

cubic polynomial cardano method.

When the discriminant is negative where the three roots are real, according to wiki, we have to use $u^3 $ and equation $(t = u - p/3u) $ to find the roots. However, cant we just use $t = u + v $ ...
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2answers
589 views

Can you help me reverse the Minimum Curvature Method?

The minimum curvature method is used in oil drilling to calculate positional data from directional data. A survey is a reading at a certain depth down the borehole that contains measured depth, ...
3
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0answers
77 views

A sequence of polynomials [duplicate]

I posted this question a while back, and I think I may not have made myself clear or shown what I got so far. So let me post this question again with more information and clarification. I have a ...
0
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1answer
25 views

Application of the Chinese Remainder Theorem for polynomials

Given the polynomials $g(t) = t$ and $h(t) = (t-3)^2 \in \mathbb{C}[t]$, I want to find the smallest (in terms of degree) polynomial $f(t) \in \mathbb{C}$ satisfying $f \equiv 0$ mod $g$ and $f \equiv ...
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2answers
20 views

What's the constant $K$ here for?

Im studying in the 10th grade and i have this problem understanding why is there the constant $K$ in the following. We are studying the relationships between the zeroes of the polynomial and we have ...
0
votes
3answers
93 views

How to prove that the roots of a quartic equation are not ALL real [closed]

Given this equation: $$x^4 + x^3 - 3x^2 + 4x - 2 = 0$$ I wanna prove that not all roots are real. How can I go about achieving this?
0
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2answers
56 views

Relation between real roots of a polynomial and real roots of its derivative

I have this question which popped in my mind while solving questions of maxima and minima. Let $f(x)$ be an $n$ degree polynomial which has $r$ real roots. Using this can we say anything about the ...
6
votes
2answers
79 views

Cyclotomic polynomials, properties.

Let $F$ be a field of characteristic prime to $n$, and let $F^a$ be an algebraic closure of $F$. Let $\zeta$ be a primitive $n$th root of unity in $F^a$. I know that the monic polynomial $\Phi_n(X)$ ...