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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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
47 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
49 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
51 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
49 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
41 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
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 ...
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 ...
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
395 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
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
69 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
34 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
300 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
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 ...
4
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5answers
343 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
142 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
33 views

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

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
105 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
711 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
votes
2answers
129 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
37 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
votes
1answer
34 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? [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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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 ...
3
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2answers
59 views

Solve $\begin{cases} x + y + z = 2 \\ 2xy - z^2 = 4 \\ \end{cases} $ for x, y, z.

It came to my mind to rewrite the expression above as $$\begin{cases} x + y = 2 - z \\ 2xy = (2 - z)^2 + 4z \\ \end{cases} $$ and see if there any restrictions on the values of the variables occur. ...
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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.
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1answer
172 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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2answers
47 views

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
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1answer
97 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 $(1-a)(1-b)(1-c)(1-d) = \frac{9}{16}$.
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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 ...
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 ...
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1answer
31 views

Factoring trinomials. [closed]

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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0answers
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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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36 views

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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1answer
60 views

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
24 views

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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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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23 views

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 [closed]

$$\frac{-c+3 a^2 b +3 a^2 d -3 a^2 f + O(3)}{-a^2 c + a^3 b + \frac{1}{2} a^3 d -2 a^3 f + O(4)}$$ 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) $\frac{1-q^{2^{1+2+\dots +6}}}{1-q}$ ...
0
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1answer
80 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?
4
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1answer
46 views

Proving that if $P(X)=a_n X^n+ \cdots+a_1 X + a_0$ has only real and simple roots then $a_{k-1}a_{k+1} \le a_k^2$

How to prove that if $P(X)=a_n X^n+ \cdots+a_1 X + a_0 \in \mathbb R[X]$ has only real and simple roots then $a_{k-1}a_{k+1} \le a_k^2$ for $1 \le k \le n-1$?
2
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3answers
32 views

Finding Linear Combination of Polynomials

I am stuck on a question involving finding the greatest common divisor of polynomials and then solving to find the linear combination of them yielding the greatest common divisor. My work thus far is ...
0
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0answers
7 views

Characterization of Multivariate Polynomials with Unique Critical Point

I would like information about $\{f\in \mathbb{C}[x_1,\ldots,x_n]:Z(\nabla(f))=\{0\}\}$. Above, the $Z$ denotes the vanishing locus of a function, i.e. the set of points where it vanishes, and ...