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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Lowest root of a quintic equation with 5 positive roots

I have a quintic equation $$ x^5-a_4 x^4+a_3 x^3-a_2 x^2+a_1 x - a_0=0 $$ with $a_n>0$ real coefficients, and I know that all 5 roots are real and positive (it is a characteristic polynomial). ...
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1answer
25 views

Maple not able to calculate Bernstein polynomial

Hope you can help me on this one. Please look at this simple Maple code: Obviously $B(1)=g(1)=4 \neq 0$. Why is Maple not able to compute this right? Am I doing something wrong? Kind regards PS: ...
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2answers
47 views

How to derive the Taylor expansion form of a polynomial expression?

I want to change this polynomial into the form $\sum_{k=0}^m a_k x^k$ $$q(x)=\sum_{k=0}^m(-1)^k\binom{2m+1}{2k+1}x^k(1-x)^{m-k}$$ I see no way to do this as I fear one might need intricate binomial ...
2
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1answer
32 views

Quartic equation or Sextic equation? And how to solve it?

In this arxiv paper (p. 11, eq. (3.2)) the authors claim that equation (3.2) is ... a quartic equation [...] which can be solved explicitly. The equation in question is \begin{equation} ...
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0answers
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Do any official publications argue that the degree of the zero polynomial should be $0$?

Usually, the degree of the zero polynomial is either left undefined, or else declared to be $-\infty.$ Anyway, I was lying in bed a few minutes ago and suddenly it occurred to me that perhaps $0$ ...
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1answer
25 views

Describing the graph of a function

For my Algebra II class one of the questions was: Describe the graph of the function $f(x) = x^3 - 18x^2 + 107x-210$. Include the $y$-intercept, $x$-intercepts, and the shape of the graph. And my ...
3
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1answer
31 views

Prove the theorem of ideal (about g.c.d)

If $p_1,\ldots,p_n$ are polynomials over a field $F$, not all of which are $0$, there is a unique monic polynomial $d$ in $F[x]$ such that (a) $d$ is in the ideal generated by $p_1, \ldots, ...
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1answer
86 views

Find $r$ in the next formula

Lets suppose I have the next values $$0, 7, 8, 5, 6$$ And the next formula $$4250 = \frac{0}{(1+r)} + \frac{7}{(1+r)^2} + \frac{8}{(1+r)^3} + \frac{5}{(1+r)^4} + \frac{6}{(1+r)^5}.$$ What is the ...
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1answer
38 views

How would you divide a polynomial by another polynomial whose power is greater than its nominator? [on hold]

I have a polynomial which is: $$\frac{(x^3-4x)}{(4x^2-4x+1)} = -10$$ Is there a way to do this? I have thought about doing long division which was not helpful...
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1answer
24 views

Functions that are “balanced” on the support of a permutation

Let $F = GF(2^n)$. Let $P(x), Q(x) \in F[x]$ be such that $P(x)$ is a permutation, while $Q(x)$ is not a permutation. For $\lambda \in F^*$ define the function $g_\lambda(x) = Tr(\lambda Q(x))$. Let ...
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2answers
30 views

Runge Phenomena and Taylor Expansion

From The Weierstrass Approximation Theorem Vs The Runge's Phenomenon: We contrast this to polynomial interpolation: this is a specific method for generating a sequence of polynomials that ...
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1answer
17 views

Notation for polynomials and equating coefficients

I am reading a paper that defines $P_k(s|t)$ as a polynomial of degree $k$ in $s$ given $t$. Does this mean that each term is of the form $f_{k}(t)s^{k}$? (What does "given $t$" mean?) The paper ...
3
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1answer
32 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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0answers
41 views

Minima and maxima of the 6th degree polynomial are not expressible in radicals.

Question: Prove that there exists a polynomial $P$ with $\deg P \geq 6$ such that the minima and maxima are not expressible in radicals. I have the following proof: the minima and maxima of a 6th ...
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3answers
78 views

Prove an equality

If $a+b+c=0$ prove that $\frac {(a^4 +b^4 +c^4)}{2}=\frac {(a^2+b^2+c^2)}{2^2}^2$ I have expanded the right side and have got this far: $a^4+b^4+c^4+2(a^2b^2+a^2c^2+b^2c^2)$ I need $a^2=b^2=c^2$ ...
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0answers
38 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
45 views

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 ...
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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 $\sum_{n=0}^{\infty} (\frac x2)^n$, ...
4
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1answer
44 views

Smooth Fano Polytopes and Hypersurfaces

This is a rather extended question, so I will try to make it as compact and readable as possible. I am trying to practice with the Macaulay2 software, in particular the polyhedra and ...
2
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1answer
47 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 ...
0
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1answer
46 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
37 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 ...
5
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4answers
93 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 ...
3
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1answer
43 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 ...
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} ...
5
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4answers
390 views

Find a polynomial from an equality

Find all polynomials for which What I have done so far: for $x=8$ we get $p(8)=0$ for $x=1$ we get $p(2)=0$ So there exists a polynomial $p(x) = (x-2)(x-8)q(x)$ This is where I get stuck. How do I ...
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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. ...
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1answer
64 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 ...
1
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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 ...
4
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3answers
297 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 ...
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0answers
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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 ...
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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 ...
5
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5answers
122 views

$f(x) =ax^6 +bx^5+cx^4+dx^3+ex^2+gx+h $ find f(7)

Problem : $f(x) =ax^6 +bx^5+cx^4+dx^3+ex^2+gx+h$ Given that : $f(1)= 1, f(2) =2 , f(3) = 3, f(4) =4, f(5)=5, f(6) =6$ find $f(7) =?$ My approach: We can put the values of $f(1) = 1$ in the ...
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0answers
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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 ...
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4answers
100 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) = ...
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4answers
691 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 ...
3
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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 ...
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2answers
35 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
32 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 ...
7
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0answers
113 views

why this polynomials is non-negative? [on hold]

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 ...
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1answer
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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.
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1answer
169 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 [on hold]

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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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; ...
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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 $ ...
3
votes
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 $(1-a)(1-b)(1-c)(1-d) = \frac{9}{16}$.
0
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1answer
22 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 ...
1
vote
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 ...
3
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0answers
76 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 ...