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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6 views

Show that $(1+a_1x+\ldots+a_rx^r)^k=1+x+x^{r+1}q(x)$

Fixed $k\ge 1$. Show that for each $r$, you can find $a_1,\ldots,a_r\in \mathbb{F}$ such that :$$(1+a_1x+\ldots+a_rx^r)^k=1+x+x^{r+1}q(x)$$ where $q(x)$ is a polynomial. Any ideas? Thanks.
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3answers
26 views

An equation over $\Bbb F_{3^k}$

Does the equation $$x^2=2=-1$$ have solutions in any extension field of $\Bbb F_3$?
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2answers
34 views

Polynomial annihilator method $y''+4y=\sin^2(2x)$

The question asks to solve the equation by this method. I know how to annihilate $\sin(2x)$ by $(D^2+4)$ however i don't know for the case $\sin^2(2x)$. Thanks!
3
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0answers
44 views

Is there any 100% sure numerical method to find all roots in a polynomial equation of degree n without fail?

Is there any 100% sure numerical method to find all roots in a polynomial equation of degree n without fail? I do not find any method which solve polynomial equation without fail.
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1answer
19 views

Clarification on how to prove polynomial representations exist for infinite series

With reference to this question, I would like a clarification of the comment given by @Ant (but someone else could answer instead). I basically have 2 questions: Is there any formal way to prove ...
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4answers
37 views

Given that$(3x-1)^7=a_7x^7+a_6x^6+a_5x^5+…+a_1x+a_0$, find $a_7+a_6+a_5+a_4+…+a_1+a_0$

Given that$$(3x-1)^7=a_7x^7+a_6x^6+a_5x^5+...+a_1x+a_0$$find $$a_7+a_6+a_5+a_4+...+a_1+a_0$$ Is there anyway to do the question without using binomial theorem and expanding the expression on the ...
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3answers
54 views

Then,what is the value of $P(0) + P(4)$?

A polynomial $P(x)$ with leading coefficient 1 of degree 4 is such that $P(\alpha)= 0$ and its roots are $1, 2$ and $3$. Then,what is the value of $P(0) + P(4)$?
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1answer
28 views

Define a matrix power by some scalars

Suppose I have an nxn matrix, for example: $$A=\begin{pmatrix}6&-2\\8&-2\end{pmatrix}$$ How is it possible to define the matrix $A^9$ using two scalars $b,c$ in R s.t.: $A^9 = bA + cI$ I ...
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2answers
64 views

What is the value of $P(6)$? [on hold]

A polynomial of degree $5$ with leading coefficient $2$ is such that $P(1) = 1, P(2) = 4, P(3) = 9, P(4) = 16, P(5) = 25$ . Then, what is the value of $P(6)$ ?
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1answer
47 views

Determine the center of ring of differential operators with coefficients in $\mathbb{C}[z_1,z_2]$

My goal is to determine what is the center of a ring $R$ generated by differential operators $z_i \frac{\partial}{\partial z_j}$ for $i,j \in \{1,2\}$ with coefficients in polynomial ring ...
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2answers
31 views

Solve $p_4(x) = x^4 −(2m + 4)x^2 + (m−2)^2 $such that $p_4$ is a product of two non-constant positive polynomials

I'm having trouble getting the starting idea for a problem I've been presented with: I need to find values for m (integer) such that the following polynomial $p_4(x) = x^4 −(2m + 4)x^2 + ...
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2answers
116 views

Problem Solving Question With Polynomials

For any polynomial $p$ with real coefficients, let $$ S(p):= \{x\in \mathbb{R} \mid p(x) \in \mathbb{Z}\} $$ Prove that if $p$, $q$ are two polynomials such that $S(p) = S(q)$, then either ...
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2answers
96 views
+150

Find all $x$ for which $g_{3n}(x)$ converges.

Let $f:\mathbb{R}\to\mathbb{R}$ be the polynomial defined by $$f(x)=x^2-x-1$$ and let $$g_n(x)=f(f(f(\cdots f(x)\cdots)))$$ be $f(x)$ applied $n$ times to itself ($g_0(x)=f(x)$, $g_1(x)=f(f(x))$, ...
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2answers
59 views

Why does an $n$th degree polynomial have at most $n-1$ turning points?

How can one explain that polynomial of degree $n$ can have up to $n-1$ turning points and $n$ intersections with the $x$-axis? If it is easier to explain, why can't a cubic function have three or ...
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1answer
49 views

Sum of roots: Vietas formulas

The equation $x^4-x^3-1=0$ has roots $\alpha, \beta, \gamma, \delta$. Find the equations with roots $\alpha^6, \beta^6, \gamma^6, \delta^6$. I was able to do this using the substitution $y=x^3$. I ...
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2answers
36 views

for which : $a_{n}$ and $b_{n}$ , $p_{n}=a_{n}x^{n+1}+b_{n}x^n+1$ to be dividing by $(x-1)²$? [on hold]

let $n \in \mathbb{n^*}$ ,set the real ensembles : $a_{n}$ and $b_{n}$ which the polynomial : $p_{n}=a_{n}x^{n+1}+b_{n}x^n+1$ dividing by $(x-1)²$ and deduce the Quotient $Q_{n}$ ? Thank you ...
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0answers
68 views

Solving systems of polynomials with an oracle

I need to solve a system of polynomials. Let the variables be $x_1, \dots, x_n$, and let the polynomials be $f_1, \dots, f_n$ Let's say we have these conditions we can already assume: there are ...
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2answers
32 views

Express the polynomial $ax^2+2hxy+2gx+2fy+by^2+c$ in matrix notation

I'm given $$\begin{bmatrix}x & y & 1\end{bmatrix}*M*\begin{bmatrix}x \\ y \\ 1\end{bmatrix}$$ where $M$ is the polynomial $ax^2+by^2+2hxy+2gx+2fy+c$ in matrix notation. Im totally stumped ...
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3answers
76 views

How to show easily that $X^4+8$ is irreducible?

Is there an easy way to show that $X^4+8$ is irreducible ? I was thinking aboute finding a $a$ such that I can use the Eisenstein criterion $(X+a)^4+8$, but I don't find a such $a$.
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1answer
88 views

Is $f(x)=0$ a polynomial function?

Is $f(x)=0$ a polynomial function? we know that constant functions are polynomials of degree zero But, does $f(x)=0$ follow this definition?
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2answers
39 views

Polynomial with positive coefficients

Consider a polynomial $P(x) = \sum_{i=1}^{n}{a_ix^{i-1}}$ in $\mathbb{C}$. Is it true that if $\{a_i\}$ are positive and not all equal, then $P(\exp(\frac{2i\pi}{n})) \neq 0$ ? Thanks
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1answer
52 views

Pattern on polynomials disguising as exponentials

Recently I've been looking at integer sequences that look like exponential at the first few terms but is actual polynomial, like these two sequences [1] [2]. And there seems to be something ...
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0answers
26 views

3 polynomial coefficient questions [on hold]

A general polynomial equation $a_nx^n+...+a_1x+a_0=0$, with $a_i$ real, cannot be solved in terms of the $a_i$ when $n>4$. Here are 3 questions I still have about this polynomials. $(n$ need not be ...
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2answers
65 views

Factor $X^7-(4+i)\in\mathbb{Q}(i)[X]$…if possible.

I think $X^7-(4+i)\in\mathbb{Q}(i)[X]$ is irreducible (simply because I don't know how to go about factoring it). Would it suffice to show that it is irreducible over $\mathbb{Z}[i]$? If so, I ...
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1answer
15 views

expressing canonical base of univariate polynomials in binomial base

Two bases are fairly standard for ${\mathbb Q}[X]$ : the canonical base $(X^j)_{j\geq 0}$ and the binomial base $(b_j(X))_{j\geq 0}$ where $b_j(X)=\binom{X}{j}=\frac{X(X-1)\ldots (X-(j-1))}{j!}$ (thus ...
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2answers
181 views

Generalisation of $(n+3)^2-(n+2)^2-(n+1)^2+n^2=4$

After seeing the neat little identity $(n+3)^2-(n+2)^2-(n+1)^2+n^2=4$ somewhere on MSE, I tried generalising this to higher consecutive powers in the form $\sum_{k=0}^a\epsilon_k(n+k)^p=C$, where $C$ ...
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3answers
59 views

prove that $f\left( x \right) = x^3 + 3x - 1$ is irreducible in $Q\left[ X \right]$

prove that $f\left( x \right) = x^3 + 3x - 1$ is irreducible in $Q\left[ X \right]$ Let $\theta$ be a root of $f(x)$ compute $\frac{1}{\theta }$; $ \left( {2 + \theta ^2 } \right)^{ - 1} $ and in ...
3
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4answers
57 views

AlgebraII factoring polynomials

equation: $2x^2 - 11x - 6$ Using the quadratic formula, I have found the zeros: $x_1 = 6, x_2 = -\frac{1}{2}$ Plug the zeros in: $2x^2 + \frac{1}{2}x - 6x - 6$ This is where I get lost. I factor ...
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0answers
32 views

Is there a injective polynomial function from $R^2$ to $R$? [duplicate]

There is an injective polynomial function from $N^2$ to $N$ (the Cantor-pairing function for example, which is of degree 2), and also one of degree 4 from $Z^2$ to $Z$. I believe the question is open ...
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0answers
29 views

How to divide two polynomials using point-value representation

I'm wondering whether there is any way to divide two polynomials represented in point-value forms ? Or Is there any tricks I can use? Point-value representation: A polynomial $f$ is evaluated at ...
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1answer
77 views
+100

How can I prove that $g(\zeta)\in\mathbb R\implies g(\zeta)=h(\zeta+\bar\zeta)$

How can I prove that if $g(X)\in \mathbb Q[X]$ and $\zeta\in\mathbb C\backslash \mathbb R$, therefore $$g(\zeta)\in\mathbb R\implies g(\zeta)=h(\zeta+\bar\zeta)$$ for a certain polynomial ...
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1answer
34 views

Transcendence degree of the ring generated by two algebraically dependent polynomials [on hold]

How to prove that If two polynomials $f$ and $g$ are algebraically dependent, then any two polynomials in the ring generated by $f$ and $g$ would be algebraically dependent?
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1answer
24 views

Basis for the space of quadratic polynomials orthogonal to those with $p(2)=p(1)$

Let $P_2[x]$ be the space of polynomials of degree less than or equal to 2. If $W = \{p ∈ P_2[x] \mid p(2) = p(1)\}$, then find a basis for $ W^⊥$ where $P_2[x]$ is equipped with an inner product ...
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1answer
33 views

Is it possible that $|z+\sum_{i\not=1} a_i z^i| <1$ for some $a_i \in \mathbb{C}$ and for all $|z|=1$?

I wonder that whether there exists a complex polynomial of the form $$ P(z)= z+\sum_{i\not=1} a_iz^i, a_i,z\in \mathbb{C},$$ (i.e. its first order term has coefficient 1) s.t. its modulus is less than ...
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1answer
28 views

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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2answers
25 views

Showing $(Tp)(x) = x^2p(x)$ is a linear map (transformation)

Define a linear map function $T: \mathcal{P}(\mathbb{R}) \to \mathcal{P}(\mathbb{R})$ where $\mathcal{P}(\mathbb{R})$ is the set of all polynomials with real-valued coefficients. Now let $T$ belong to ...
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0answers
60 views

Replacing numbers by roots of quadratic

We have $10$ numbers in the interval $(0,1)$, not necessarily distinct. At any moment, we can choose two of them, $a$ and $b$. If the quadratic $x^2-ax+b$ has two (possibly identical) real roots, we ...
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2answers
33 views

Simplification ideas

Looking for a neat simplification idea to be able to solve for $x$ analytically in the expression below: $$S=k\tan x-Bk^2\frac{1}{\cos^2x}$$ where $\{S,k,B\}\neq0$ and $\in \mathbb{R}^+.$ Of ...
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0answers
11 views

Coefficients of Lagrange resolvent

I'm trying to make sense of some things I read about Galois theory. Let $p$ be a monic polynomial of degree $n$ with known coefficients $a_i$ and unknown roots $x_i$: \begin{alignat*}{2} p(X) &= ...
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5answers
269 views

Sum of roots: Vieta's Formula

The roots of the equation $x^4-5x^2+2x-1=0$ are $\alpha, \beta, \gamma, \delta$. Let $S_n=\alpha^n +\beta^n+\gamma^n+\delta^n$ Show that $S_{n+4}-5S_{n+2}+2S_{n+1}-S_{n}=0$ I have no idea how to ...
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2answers
61 views

How do I take the 100th derivative of a polynomial [closed]

How could I find $$f^{100}(x)$$ for $$f(x)=2x^{100}-7x^{80}+15x^{60}-27x^{40}-18x^{20}+300$$
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2answers
119 views

How to create a computationally cheap function passing through given points?

I am trying to develop a function which goes through the follow points. The function will be calculated on a microprocessor which has 20 mHz. List of given points: ...
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0answers
43 views

Canonical algebra isomorphism $k[D(f)]\cong k[S_0,\dots,S_n]_{(f)}$?

Here's a common set up. Suppose you have $f\in k[S_0,S_1,\dots,S_n]$ is a homogeneous polynomial with $\deg(f)=d$, over some closed field $k$. Let $D(f)$ be the principal open set of $f$ in projective ...
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0answers
14 views

Multitangent to a polinomial function

I'm trying to build some exercises on tangents of functions for beginner students in mathematical analysis. In particular I would like to suggest the study of polynomial functions $ y = p (x) $ of ...
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1answer
39 views

Diophantine equation : two products of linear factors differ by a constant

Recently, I was asked the following question by a friend : find all $a,b,c,a',b',c',k \in {\mathbb Z}$ with $k\neq 0$ such that the identity $$ (X-a)(X-b)(X-c)+k=(X-a')(X-b')(X-c') $$ holds in ...
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1answer
30 views

Having trouble combining Weierstrass approximation theorem and the infinite sequence of holomorphic functions

The Weierstrass approximation theorem says that all continuous functions on $[0,1]$ can be approximated uniformly by polynomials. Trying to facilitate the digestion of the fatty Christmas food, I ...
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3answers
67 views

Sum of Coefficients in a Polynomial

Find the sum of the coefficients of the terms in the expansion of $(2x+3y-3z)^7$. I know how to do this for binomials, but I was not able to apply the same logic to a trinomial. I believe my other ...
0
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0answers
36 views

What bounds can we put on the largest root of a polynomial?

For a polynomial $p(x)=x^{n+1}+a_{n} x^{n} + \cdots + a_1$ with roots $|x_1| < \cdots < |x_n|$ can we find relatively simple function $M(a_1, \dots, a_n)$ such that $$|x_i| \leq M(a_1, \dots, ...
1
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1answer
29 views

Sum of Coefficients and Number of Terms in Trinomials and Quadrinomials

I already know how to find the sum of coefficients in a binomial, but how do you do it for a trinomial/quadrinomial (after like terms are added)? Example Problem: $(wa+xb+yc+zd)^n$ (all variables are ...
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1answer
35 views

Find a probability of $n$ event happening from $m$ types

The question is: to find a sum $$ S=\sum\limits_{n_1+n_2+\ldots+n_m = n,\ n_i=0,1,\ldots,n} p_1^{n_1}p_2^{n_2}\cdots p_m^{n_m}, $$ where $p_i\in[0,1]$. UPDATE. This issue has no probabalistic ...