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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102 views

Find the coefficient of $x^{30}$.

Find the coefficient of $x^{30}$ in the given polynomial $$ \left(1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^{10}+x^{11}+x^{12}\right)^5 $$ I don't know how to solve problems with such high degree.
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3answers
30 views

What is that function? Polynomial?

Is it a polynomial or rational polynomial or else? $y = \dfrac{a}{x^4} + \dfrac {b}{x^2} + c$ I need to fit a curve to a discrete data of that form, so I need to know what fitting to use.
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0answers
27 views

coefficients of polynomial and binomial expressions

Let us say we are given a polynomial p(x)=$\sum_k a_k x^k$. In order to find $\sum_k a_k$ we simply need to evaluate p(1), and similarly there are many other tricks. Is there any trick to evaluate ...
1
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1answer
46 views

An approach to proving that $\Bbb{Q}[x,y]/(x^3-y^2)$ is isomorphic to $\Bbb{Q}[t^2,t^3]$

I have to prove that $\Bbb{Q}[x,y]/(x^3-y^2)$ is isomorphic to $\Bbb{Q}[t^2,t^3]$. My approach: Let us consider $t^2$ and $t^3$ as separate variables $x$ and $y$. The relations that hold for them ...
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3answers
52 views

Prove $\lim_\limits{x\to\infty}\dfrac{P_k(x)}{P_{k+1}(x)}=0$ [closed]

Prove $\lim_\limits{x\to\infty}\dfrac{P_k(x)}{P_{k+1}(x)}=0$ by limits. $P_k(x)$ is defined as a polynomial of degree $k$.
3
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1answer
63 views

Is there a formula for the closed form for $ \displaystyle \sum_{r=1}^\infty \frac{\sum_{k=1}^r k^n}{r!}$ for any positive integer $n$?

Is there a formula for the closed form for $ \displaystyle \sum_{r=1}^\infty \frac{\sum_{k=1}^r k^n}{r!}$ for any positive integer $n$? I tried Faulhaber's formula and Bell number but couldn't ...
0
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3answers
21 views

Consider the equation [x^5+x=10] show that i)the equation has only 1 real root ii)this root lies between 1 & 2 iii)the root must be irrational [closed]

This equation obviously has 5 roots..If they are considered as a,b,c,d,e then a+b+c+d+e=0,abcde=-10..but what next?If i procced through contradiction will it help?
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0answers
29 views

Given $f=x^4+x+1 \in \mathbb Z_{2}[x]$ is primitive, write down an $m$-sequence ${a_n}$ associated to $f$

I'm not sure how to solve this question exactly. I know that the period will be 15 but I don't know how to construct the $m$-sequence.
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0answers
60 views

Solving quartic equation? (Cardano/Ferrari)

today I've written a little Code-Snippet that is based upon the steps that are mentionned in this wikipedia-Article to solve a general quartic polynom. Here's my matlab-implementation: ...
3
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0answers
91 views

Cardinomials: Like cardinalities, but polynomial valued

I want to see if this notion is known (or if it makes sense). Let $F$ be a field. Let $A$ be a finite dimensional commutative unital algebra over $F$. Let $X_1$, $X_2 \in A$ etc. be such that their ...
0
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1answer
37 views

general theorem on roots of a polynomial needed to show it's identically zero.

Polynomial degree k, one variable, if it's zero at k+1 values, then it's identically zero. Can someone point me to a proof of this? I know derivatives at points can count as these roots (if k-degree ...
3
votes
2answers
72 views

Prove: if $a$ and $b$ are algebraic, then $a + b$, $a - b$ and ab are also algebraic

I have to prove the following: If $a, b \in \mathbb{C}$ and are both algebraic over $\mathbb{Z}$, then: $a + b$ is algebraic over $\mathbb{Z}$ $a - b$ is algebraic over $\mathbb{Z}$ $ab$ is ...
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votes
5answers
49 views

Cubic Equation. (Factorisation)

I'm given this question, factorise $4x^3-7x-3$. Is this answer acceptable? $(x+\frac{1}{2})(x-\frac{3}{2})(x+1)$.
4
votes
1answer
65 views

Find $n$ such that $x^2 + x + 1$ is a factor of $(x+1)^n - x^n - 1$.

I have to find the form of n i.e. whether n is even or odd and whether it is multiple of 2 or 3 such that: $x^2 + x + 1$ is a factor of $(x+1)^n - x^n - 1$. What I tried: $x^2 + x + 1 = (x + 1)^2 - ...
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1answer
13 views

Solving the Cubic Equation (using Lagrange Resolvents)

This is from my textbook. I am having trouble working out the calculations that the author skips over. So we start with the polynomial $\ X^3 - aX^2 + bX -c$ with zeros $x_1,x_2,x_3$. Then we define ...
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1answer
14 views

How to find representation of polynomial w.r.t different basis

Let $B$ be the basis of the vector space of polynomials of degree less than or equal to 2. $B = \{1, t-1,(t-1)^2\}$. Let $u = 2t^2-5t+6$. How do you find $u_b$, the coordinate vector of $u$ relative ...
0
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0answers
22 views

$S_k(x+y)-S_k(x)-S_k(y)$ where $S_k$ is symmetric polynomial

Let $S_k$ be the $k$-th symmetric polynomial of $n$-variable. How can I rewrite $$S_k(x+y)-S_k(x)-S_k(y)$$ by just using $x,y,S_1,S_2,\cdots S_{k-1}$ where $x=(x_1,x_2,\cdots,x_n)$ and ...
0
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1answer
21 views

if $f(n+1)-f(n)=P(n)$, exist a polynomial $Q(x)$ such that for all $n \in \mathbb{Z}$ : $Q(n)=f(n)$

Let $f:\mathbb{Z} \to \mathbb{Z}$ such that, exist a polynomial $P(x)$: $$f(n+1)-f(n)=P(n)$$ for all $n \in \mathbb{Z}$ Prove that exist a polynomial $Q(x)$ such that for all $n \in ...
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0answers
12 views

Recursive relationship for Peano Baker Series

The Peano Baker Series is a integral has the following form $$\varPhi(h,0)=I+\intop_0^h G(t_{1}) \, dt_1 + \intop_0^h G(t_1) \intop_0^{t_{1}} G(t_2) \, dt_2 \, dt_1 + \intop_0^h G(t_1) ...
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0answers
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+50

Is the given binomial sum almost everywhere negative as $K\to\infty$?

The binomial sum is as follows: $$\mathcal {L}^K(\theta)= \sum_{i=\lceil{K/2}\rceil}^K \binom{K}{i}\theta^i\left((1-\theta)^{K-i}-\frac{1}{2}(1-\theta)^{-K}(1-2\theta)^{K-i}\right)$$ which can also ...
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0answers
32 views

seeking for Newton's like inequalities as sufficient condition for polynomial to have only real zeros

For polynomial $P_n(x)=\sum_{k=0}^n a_k x^k, a_k>0$, it is known that a necessary condition for $P_n(x)$ to have only real zeros is that Newton's inequality holds: ...
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1answer
28 views

An obstacle encountered in a proof of the existence of a best approximating polynomial of degree $\leq n$

Let $n \in \{0, 1, 2, \dots\}$, let $a, b \in \mathbb{R}$ be such that $a < b$ and let $f \in \mathcal{C}[a, b]$ be a real function that is continuous on the non-degenerate, compact interval $[a, ...
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1answer
39 views

Equation that defines multi-dimensional polynomial

In two-dimensions a complete n-th degree polynomial is given by $P_n(x,y) = \sum_{k=0}^{n}\alpha_kx^iy^j \qquad i+j \leq k \qquad (1)$ . However, now I am dealing with the following two-dimensional ...
2
votes
1answer
66 views

Ordered Pairs of Polynomials

Professor proposed this problem to the class today. Suppose we had $P_1(x), P_2(x) \in \mathbb{Z[x]}$, $n, a \in \mathbb{Z}$. How many ordered pairs exist such that ...
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1answer
40 views

I have to show $p=p(x-\lambda)$ if and only if they have the same zeros in $F$

Suppose $F$ is a field, $|F|\geq n \geq 2$. Given $\lambda \in F$ I know $p,p(x-\lambda)\in F[x]$ are irreducible monic polynomials. I have to show $p=p(x-\lambda)$ if and only if they have the same ...
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1answer
52 views

Taylor polynomial for an integral

This is the first time encountering a Taylor expansion along with an integral, so I am wondering how I should proceed. Question: $Consider \space the \space function$ $$F(x) = ...
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0answers
21 views

Zero Homogeneous Polynomials

Let $f:\mathbb{R}^n\rightarrow \mathbb{R}$ be a homogeneous polynomial of degree n. Is it true that if $\forall x,f(x)=0$, then the coefficients of $f$ are all zero?
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1answer
20 views

Write down basis for the set of all polynomials $f(x)$ of degree at most 5 such that $f(2) = 0$.

Write down basis for the set of all polynomials $f(x)$ of degree at most 5 such that $f(2) = 0$. I know there are lots of answers you could write, but would this be correct: $\{(x-2)^5, (x-2)^4, ...
0
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0answers
14 views

Polynomial Division for crc

I did this question by just using the xor long division of the binary, but my teacher said he doesn't want it done that way, but want me to use polynomial Division. I have no clue how to do this, and ...
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1answer
31 views

Find all polynomials such that $P(A)\subset U$ for a countable subset of the unit circle $U$

I recently answered a question, in which I proved that If a polynomial fixes the unit circle then $P$ is a monomial (a classical result),i,e: $$\forall P\in \Bbb C[X]\ \ \ \ (\forall z\in \Bbb C \ \ ...
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1answer
161 views

Help me understand Gröbner basis result please

I'm practicing a bit with Gröbner bases but I'm not understanding the following result I obtain from Mathematica: ...
2
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1answer
30 views

If $|\det(A+zB)|=1$ for any $z\in \mathbb{C}$ such that $|z|=1$, then $A^n=O_n$.

Let $A,B\in \mathcal{M}_n(\mathbb{C})$ such that $AB=BA$ and $\det > B\neq 0$. a) If $|\det(A+zB)|=1$ for any $z\in \mathbb{C}$ such that $|z|=1$, then $A^n=O_n$. b) Is the ...
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0answers
9 views

Writing a series of polynomial equations of certain degree from a sequence of binary bits using Magma

How do I write a series of polynomial equations of a specified degree from a sequence of binary bits using Magma. So far, I have the following code for converting a decimal sequence to binary. ...
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2answers
56 views

Polynomials mod prime $p$

The problem is $5m^2+m+4 \equiv 0\pmod 7$. I am supposed to first convert it to a quadratic whose first coefficient is $1$. But the polynomial cannot be factored, so I am unsure as to how to do ...
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0answers
14 views

Let $P(x),Q(x) \in \mathbb{Z}[x]$ such that, exist $a,b \in \mathbb{Z}^+$ and $a<b$: $P(a)=Q(a)$ and $P(b)=Q(b)$

Let $P(x),Q(x) \in \mathbb{Z}[x]$ such that, exist $a,b \in \mathbb{Z}^+$ and $a<b$: $P(a)=Q(a)$ and $P(b)=Q(b)$ Prove that $P \equiv Q$
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1answer
23 views

For any $n \in \mathbb{Z^+}$ Not extis $P(x) \in \mathbb{R}[x]$ with coefficients in $B$ and all roots of $P(x)$ in $A$

Problem: Let $A=\{a_1,a_2,..,a_m\}$ and $B=\{b_1,b_2,...,b_p\}$ where $a_1,a_2,...,a_m,b_1,b_2,...,b_p \in \mathbb{R}$ Prove that , the following statements is bad : for any $n \in ...
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1answer
55 views

Help required! Polynomials

Let $D(p) = p^{20} - p^{18} - p^{16} - \dots - p^2 - 2$ Prove that the sum of fourth powers of all the real roots of $D(p) = 8.$ Please help.
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1answer
17 views

Limit at $\infty$ of a polynomial multiplied by a negative exponential

I am trying to show $\int_0^{\infty} x^2 e^{-2 x} dx = 1/4 $ Integration by parts gets the indefinite integral $$\int x^2 e^{-2 x} dx = \frac{-1}{4} e^{-2 x} (2 x^2+2 x+1)+constant$$ In order to ...
2
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1answer
41 views

Determine the units of the ring $A= \mathbb Z[X]/(X^3)$ and the structure of the group $A^*$ [duplicate]

Determine the units of the ring $A= \mathbb Z[X]/(X^3)$ and the structure of the group $A^*$ I've only managed to show that the free coefficient of any unit in $A$ is a unit in $\mathbb Z$.
2
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4answers
59 views

Solve the following integral: $ \int \frac{x^2}{x^2+x-2} dx $

Solve the integral: $ \int \frac{x^2}{x^2+x-2} dx $ I was hoping that writing it in the form $ \int 1 - \frac{x-2}{x^2+x-2} dx $ would help but I'm still not getting anywhere. In the example it was ...
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0answers
16 views

Are there Karnaugh maps over other algebras?

Karnaugh maps are a useful way to minimize or factorize polynomial expressions in Boolean algebra by considering the smallest combinations of logical "subcomponents" of an expression, whose sum is ...
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2answers
44 views

$x^3+ (5m+1)x+ 5n+1$ is irreducible over $\Bbb Z$

How to prove that the polynomial: $x^3+ (5m+1)x+ 5n+1$ is irreducible over the set of integers for any integers $m$ and $n$? I was trying to put $x= y+p$ for some integer $p$ so that I could apply ...
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1answer
29 views

Lagrange Interpolation Polynomial Degree N [closed]

I want a Lagrange Interpolation formulae/code/online calculator which determines a apolynomail of degree n passing through given points.
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2answers
59 views

How should you go about simplying cubic polynomial: $y(x) = x^3+12x^2+21x+10$

Claim: $$y(x) = x^3+12x^2+21x+10$$ Can be factored into $$(x+1)^2(x+10)$$ But what is the quickest way to see this?
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1answer
31 views

What are the steps to function design?

So I'm trying to write a program, and I want to use math functions to help it. In this example, I'm trying to change the color of a line based on the position of each pixel on the line. Anyway, I ...
0
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2answers
42 views

Same roots, same polynomial? How to prove characteristic polynomial of $AB = BA$?

I'm giving a (simple) proof that the characteristic polynomial of $AB$ = characteristic polynomial of $BA$ (without using the fact that $AB$ and $BA$ are similar). $det(AB) = det(A)det(B) = ...
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1answer
38 views

How to prove whether $x^{2}+y^{2}+1$ is irreducible over $\mathbb{C}$ or not?

Let's consider a 2-variable polynomial $f(x, y)= x^{2}+y^{2}+1$. It can be established that it's irreducible over $\mathbb{R}$. For example, if it's irreducible over $\mathbb{R}$ as a polynomial of ...
0
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0answers
30 views

Lagrange interpolation given a list of points

I have to calculate a value in which I use Lagrange interpolate to calculate numerator and denominator individually. On dividing the interpolated numerator and denominator I don't get the required ...
0
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0answers
7 views

A question related Kharitonov's Theorem for Hurwitz stable interval polynomials

Definition: An interval polynomial is the family of all polynomials $$p(s)= a_0 + a_1 s^1 + a_2 s^2 + ... + a_n s^n\tag{1}$$ where each coefficient $a_i \in \mathbb{R}$ can take any value in the ...
4
votes
4answers
115 views

$tr(A)=tr(A^{2})= \ldots = tr(A^{n})=0$ implies $A$ nilpotency

Let's consider a $n \times n$ matrix and the sequence of traces $tr(A)=tr(A^{2})= \ldots = tr(A^{n})=0$. How to prove that $A$ is a nilpotent matrix (a matrix so that $A^{k} \times u = 0$ for all $u ...