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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Notation for polynomials and equating coefficients

I am reading a paper where they define $P_k(s_1,s_2|t)$ as a polynomial of degree $k$ in $s_1$ and $s_2$ given $t$. What does it mean "given $t$"? (I was thinking that each term looks like ...
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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 ...
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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
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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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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 ...
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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 ...
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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 $\sum_{n=0}^{\infty} (\frac x2)^n$, ...
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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 ...
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1answer
45 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 ...
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1answer
44 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 ...
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1answer
32 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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4answers
88 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 ...
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1answer
41 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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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} ...
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384 views

Find a polynomial from an equality

Find all polynomials for which $$(x-8)p(2x)=8(x-1)p(x)$$ 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 ...
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1answer
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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
62 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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$x^p -x-1$ 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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295 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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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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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 ...
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$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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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
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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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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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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 ...
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1answer
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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 ...
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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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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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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
168 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
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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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Polynomial problem solving: find the values of $a$ and $b$. [on hold]

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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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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1answer
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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 $(1-a)(1-b)(1-c)(1-d) = \frac{9}{16}$.
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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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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 ...
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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 ...
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1answer
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Factoring trinomials. [on hold]

A student factored $m^2 + 12mn + 144n^2$ as shown. I know that since $m^2$ squares = $m^4$ and $144n^2$ squared = $12n$, the first and third terms of the trinomial are perfect squares. This means ...
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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 ...
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Sum of Bell Polynomials of the Second Kind

A problem of interest that has come up for me recently is solving the following $$\frac{d^{n}}{dt^{n}}e^{g(t)}$$ There is a formula for a general $n$-th order derivative of a composition as shown ...
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Confusions about Cardano's method to solve cubic

Here are some questions that I don't regarding to the Cardano's method. 1, Is $q^3/27 + p^2/4$ the discriminant for cubic? 2, what is casus irreducibilis? Does it mean using radicals to represent ...
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Maths cubic equation discriminant…

So I am now researching for the cardano method and I do not understand where did the cubic discriminant come from..... It must be from the cardano method..... Also in this video 2 min 21 sec ...
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1answer
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Dividing polynomials on all terms

Can you please explain why the following operation is wrong: Expression: $2 [(x+8) + x] = 208$ Operation: To remove the $ 2$, divide both sides by $2$. The $ 2$ cancels out on the left. This leaves ...
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74 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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How to calculate the $k$-dimension of a subspace of a polynomial ring?

Let $k$ be an infinite field and $R:=k[x_1,...,x_n]$ the polynomial ring in $n$ indeterminates. Why is the $k$-dimension of $U$ given by $\begin{pmatrix} n+m-1 \\ m\end{pmatrix}$, when $U$ is the ...
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How should this power series be solved [on hold]

$$\frac{-c+3 a^2 b +3 a^2 d -3 a^2 f + O(3)}{-a^2 c + a^3 b + \frac{1}{2} a^3 d -2 a^3 f + O(4)}$$ then its answer is required up to $ O(a)$ here in both equations a is a variable and $ b,c,d,f $ are ...
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5answers
83 views

The expression $(1+q)(1+q^2)(1+q^4)(1+q^8)(1+q^{16})(1+q^{32})(1+q^{64})$ where $q\ne 1$, equals

The expression $(1+q)(1+q^2)(1+q^4)(1+q^8)(1+q^{16})(1+q^{32})(1+q^{64})$ where $q\ne 1$, equals (A) $\frac{1-q^{128}}{1-q}$ (B) $\frac{1-q^{64}}{1-q}$ (C) $\frac{1-q^{2^{1+2+\dots +6}}}{1-q}$ ...
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1answer
77 views

Factorizing a cubic polynomial

This is the result of determinant evaluation: $$p(x) = (x-3)((x-1)(x-2)-1)+1$$ How can I factor this polynomial?