This tag is used for both basic and advanced questions on polynomials in any number of variables. Including, but not limited to: solving for roots, factoring, checking for irreducibility. This tag is rarely used as the only tag for a question.

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5
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2answers
134 views

Newton's identities over finite fields

The Newton identities (including over finite fields) are given by $$ ke_k = \sum_{i=1}^k (-1)^{i-1} e_{k-i}p_i, $$ where the $e_k$ is the $k$-th elementary symmetric polynomials and the $p_k$ is the $...
0
votes
2answers
73 views

Identify the roots for the equation: $(x-2)(x-6i)(x+6i)= 0$

$$(x-2)(x-6i)(x+6i)= 0$$I'm not exactly sure if I fully understand this question. If I'm not mistaken, I managed to get the results: $r_1=2, r_2=6i$, and $r_3=-6i$. However I'm not entirely certain ...
2
votes
2answers
110 views

Write a polynomial equation of the smallest degree with roots $3$, $4i$, and $-4i$.

I already have the answer to this question (it's an example). However, I am still confused with steps 2-3. How does $x^2+4ix-4ix-16i^2$ simplify to $x^2-16$? $(x-3)(x-4i)(x+4i)=0$ $(x-3)(x^2+4ix-4ix-...
0
votes
1answer
22 views

Question about polynomial ring and coefficients

Let $R=k[x_0,...,x_n]$ be the polynomial ring in $n+1$ variables and let $F=c_1f_1+ \cdots +c_kf_k \in R$ with $c_i \in k$. Is it possible to multiply $F$ with some element from $k$ such that the ...
2
votes
1answer
52 views

If $\alpha$ is a root of $f(t) = t^n + a_{n-1}t^{n-1} + \cdots + a_0$, then $|\alpha| \leq n \max_i |a_i|$

Let $f(t) = t^n + a_{n-1}t^{n-1} + \cdots + a_0$. Let $\alpha$ be a root of $f$. Then show that $\alpha \leq n \max_{i} |a_i|$. I could only figure it out for the special case where $|a_i| < 1$ ...
1
vote
1answer
46 views

Question about $q$-expansion principle for modular forms

I am reading Zagier's lectures "The 1-2-3 of Modular Forms". Before formulate the question let me introduce some notations. Let $z$ be the coordinate on the upper-half plane $\mathbb{H}$ and $q=\...
3
votes
1answer
77 views

Properties of distribution of zeros of polynomial

Polynomial $p_n(z) = (1 + \frac{z}{n})^n - 1$ has a property that all its zeros lie on the circle of radius $n$. It is easy to see because $$\frac{z}{n} = e^{\frac{i2\pi k}{n}} - 1$$ So we can "fit" ...
0
votes
1answer
67 views

Can we find sufficient conditions in which this equation have only three distinct real roots

Let us consider the polynomial equation: $$ξ₁x⁸+ξ₂x⁷+ξ₃ x⁶+ξ₄ x⁵+ξ₅ x⁴+ξ₆ x³+ξ₇ x²+(ξ₈-1) x+ξ₉ =0$$ where $ξ_{i}$ are real coefficients. My question is: Can we find sufficient conditions in ...
8
votes
5answers
420 views

Minimal Polynomial of $\sqrt{2}+\sqrt{3}+\sqrt{5}$

To find the above minimal polynomial, let $$x=\sqrt{2}+\sqrt{3}+\sqrt{5}$$ $$x^2=10+2\sqrt{6}+2\sqrt{10}+2\sqrt{15}$$ Subtracting 10 and squaring gives $$x^4-20x^2+100=4(31+2\sqrt{60}+2\sqrt{90}+2\...
3
votes
2answers
66 views

Why are there at most $n-1$ positive roots for polynomials with prime powers?

I was attempting to solve this old contest math problem posted Show that a matrix has positive determinant yesterday and I realize that I don't even know why the hint provided is true. From that post:...
0
votes
1answer
45 views

$f(x)\in K[x]$ implies $\deg(f)\mid [E:F]$

Let $F$ be a field, $f(x)\in F[X]$ irreducible, $n$ the degree of $f(x)$, and $E/F$ the splitting field of $f(x)$. I want to prove that $n\mid [E:F]$. I try this by induction. $n=1$ is trivial. ...
1
vote
1answer
31 views

Barbeau's Polynomials: Quadratic Polynomials, 1.2.2

I've verified $(a)$ by expanding the $RHS$. I've partially verified $(b)$ doing the following: $$\begin{eqnarray*} {p(t)}&\stackrel{?}{=}&{p(t)-p(r)}&\stackrel{?}{=}&{(t-r)(at+...
2
votes
1answer
49 views

Extended Liouville Polynomial Inequality

I am trying to show that the noted inequality holds without using the Euclidean Algorithm. This appears in Complex Analysis by Newman and Bak on the top of page 63 (remark 1, my paraphrase): If $\...
3
votes
1answer
34 views

Can a polynomial equation always be manipulated to give a recurrence formula?

Let $p(x)$ be a real (or maybe complex) polynomial. Suppose we wish to (numerically) solve $p(x) = 0$. This can be done for example with Newton's method of course, but I was thinking about if you "...
2
votes
2answers
150 views

Find $p$, such that $\frac1{20} = (1 - p)^{19}p$

I need help to solve for $p$, where $p$ is a probability, i.e. it lies in the interval $[0,1]$. $$\frac1{20} = (1 - p)^{19}p.$$ How would one solve for $p$? Thnx
2
votes
1answer
52 views

polynomial with nonzero coefficients at prime degree terms

Let $P(x)$ be a polynomial with integer coefficients. Show that there is a non-zero polynomial $Q(x)$ with integer coefficients, such that the product $$P(x)Q(x)=\sum_{k\ge 0}a_k x^k$$ has only ...
6
votes
6answers
267 views

Prove that $f=(x+i)^{10}+(x-i)^{10}$ have all real roots

We have $f=(x+i)^{10}+(x-i)^{10}$ and we need to prove that $f$ have all the roots in $\mathbb{R}$. Here is all my steps: Suppose that $z\in\mathbb{R}$ is a root of $f\Rightarrow (z+i)^{10}+(z-i)^{...
1
vote
0answers
45 views

Factorizing a Polynomial over the Integers

What are the most efficient algorithms to factorize a polynomial over integers, knowing that it has only integer roots? I googled around a lot, but most of the work seems to be around Finite fields. ...
2
votes
1answer
146 views

Solutions of sixth order polynomial equations

Do you know a way to solve exactly a general sixth order polynomial equation: $x^{6}+a_{5}x^{5}+a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}=0$ ? According to this link, it is possible to solve it ...
2
votes
1answer
53 views

Finding roots of an irreducible polynomial in a ring that is not a domain.

Let $A$ be a commutative ring, and $p\in A[X]$ a polynomial of degree $d>0$. If $A$ is an integral domain, we can find a ring $B$ such that $A\subseteq B$ and $p$ has a root in $B$. For example ...
12
votes
4answers
1k views

Algorithm(s) for computing an elementary symmetric polynomial

I've run into an application where I need to compute a bunch of elementary symmetric polynomials. It is trivial to compute a sum or product of quantities, of course, so my concern is with computing ...
3
votes
2answers
130 views

Transform a polynomial so that positive roots are shifted right and negative roots are shifted left

I'm trying to figure out if it is possible to shift the roots of a polynomial outward, instead of to the left or right. Its relatively simple to shift all the solutions in one direction by ...
0
votes
0answers
51 views

Product of Polynomials of Binary Variables: Linearization

I have the following term (in the context of mathematical programming): $$\prod_{p = 1}^P [1 - z_p(1 - \lambda_p)]$$ where $\lambda_p \in [0,1]$ is a parameter and $z_p \in \{0,1\}$ is a binary ...
3
votes
1answer
67 views

Why is the solution to $y' = y^n$ always in polynomial form EXCEPT when $n = 1$?

Could someone explain (intuition-wise) why the differential equation $$y' = y^n$$ for $n \in \mathbb{N}$ seems to always some kind of polynomial solution (or a ratio of polynomials, etc.) except ...
3
votes
0answers
172 views

Properties of polynomials that are polynomial conditions on the coefficients

There are many occasions where we can check whether a (set of) polynomial(s) $f_i$ satisfies certain properties, simply by evaluating a fixed polynomial on the coefficients of the $f_i$. Many times, ...
2
votes
2answers
94 views

Polynomial in several variables over $GF(2)$

Can anyone please explain how this Lemma has been proved? Lemma: Let $f$ be a nonzero polynomial in variables $x_1,\ldots,x_n$ over $GF(2)$, and let $d$ be the maximum degree of $f$ with respect to ...
1
vote
3answers
99 views

Find all primes $p>2$ for which $x^2+x+1$ is irreducible in $\mathbb{F}_p[x]$ [duplicate]

Find all primes $p>2$ for which $x^2+x+1$ is irreducible in $\mathbb{F}_p[x]$ Attempt. Since $x^2+x+1$ is of degree 2, it is reducible iff it has a root in $\mathbb{F}_p$. It has a root in $\...
0
votes
2answers
6k views

Graphing: Given two points on a graph, find the logarithmic function that passes through both.

Is there such a method to do this? I would like to come up with a logarithmic function (a graph that looks like a square root graph) that passes through two given points. Haven't had any luck in ...
2
votes
2answers
1k views

Is there a rule to the terms of a falling factorial?

$\require{cancel}$I discovered that $n!=\xcancel{(n)_{n-1}}n^{\underline{n-1}}=n(n-1)(n-2)\cdots(3)(2)$. I have expanded a few examples: $$2!=\xcancel{(2)_1}2^{\underline{1}}=2\\ 3!=\xcancel{(3)_2}3^{...
9
votes
1answer
415 views

Reducible polynomials in $\mathbb{Z}[X]$

Let $(a_n)_{n\geq 1}$ be a strictly increasing sequence of integers and $k$ an integer different from $0$. There exists among the polynomials $$ (X-a_1)(X-a_2)\cdots(X-a_n)+k,\ n\geq 1 $$ only a ...
-1
votes
1answer
43 views

Deriving a formula from a set of points

I have a set of values that are increasing non uniformly. I can figure out an equation for a constant increase, but this is not linear. How do I determine a formula for these points? 2179, 2197, 2247,...
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0answers
14 views

Proof remainder of polynomial division GF(2) can be calculated by LSFR

I have been reading that CRC, which is the calculation of the remainder of $x/P(x)$ in GF(2) can be implemented with a Linear Shift Feedback Register. However, I can't find the proof for this, or ...
7
votes
1answer
367 views

Factoring polynomials with prime discriminant

I was busy doing a homework exercise in which I had to compute the discriminant $\Delta(f)$ of the polynomial $$f(X) = X^4+X^2+X+1$$ which turned out to be the prime $257$. Subsequently, I was asked ...
2
votes
1answer
70 views

Irreducibility of polynomials $x^{2^{n}}+1$

I would like to if the polynomials of the form $x^{2^{n}}+1$ are irreducible over $\mathbb{Q}$ and in that case if there is some "easy" proof for that (where easy means not using a big theory like ...
1
vote
0answers
44 views

Why is polynomial convolution equivalent to multiplication in F[x]/(xn−1)?

Why is polynomial convolution equivalent to multiplication in $F[x]/(x^n-1)$? From this, I still can not understand how to get this $$ \begin{align} &f*g +(x^n-1)\sum_{k=0}^{n-1}\sum_{i+j=k+n}...
0
votes
3answers
155 views

Polynomials - Remainder thereom + factor thereom

Write $p(x) = x^4 + 4x^3 - 14x^2 - 36x + 45$ as a product of its factors. My solution so far: $p(3) = 0$ therefore it's a factor $$ (x^4 + 4x^3 - 14x^2 - 36x + 45)\big/(x-3) ~=~ x^3 + 7x^2 + 7x - 15 ...
1
vote
1answer
57 views

Proving that f(z), bounded above by C|z|^n for some constant C, is a polynomial of degree at most n:

I realized some crucial mistakes I had made when posting this question, so here is an edited version that is less wordy and hopefully a better question: From using the Cauchy Integral Formula, and ...
1
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1answer
54 views

Are there any entire functions that have a finite number of non-zero terms in their Taylor expansions?

Besides polynomials, of course. I am pretty sure the answer will be "no", but can we actually prove this? Thanks,
1
vote
3answers
61 views

Why is $(XY-1)$ contained in $(X-a, Y-b)$ with $ab=1$?

This is probably a very trivial question, so I apologize in advance. Let $K$ be an algebraically closed field and $R=K[X,Y]$ the polynomial ring in two variables. I want to show that every ideal $...
5
votes
3answers
255 views

Factors in a cubic equation

I have no idea how to go about this. Any Hint? Suppose that $(x-3)$ is a factor of $$kx^3 - 6x^2 + 2kx - 12.$$ Solve for $k$.
2
votes
2answers
39 views

Partial Fraction Decomposition of a Polynomial division

Question :Write $$\frac{x^5}{(x^2+1 )(x+1)^2}$$ as a sum of partial fraction What I've tried is to do polynomial long division twice to reduce the degree of numerator to be smaller than denominator ...
2
votes
1answer
28 views

Possible division by zero when repeatedly factoring and canceling (x-r) and evaluating the resulting polynomial at x=r…

The following proof is from 'Two Proofs of the Existence and Uniqueness of the Partial Fraction Decomposition', page 4. Lemma 2.1. Let $h(x), g(x) \in R[x]$ (real polynomials), $k$ a positive ...
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vote
2answers
2k views

Why is polynomial convolution equivalent to multiplication in $F[x]/(x^n-1)$?

(Context: polynomial multiplication using DFT/FFT) Let $f = \sum\limits_{i=0}^{n-1} f_i x^i$ and $g = \sum\limits_{j=0}^{n-1} g_j x^j$ be polynomials in $F[x]$ for some field $F.$ The convolution of $...
1
vote
1answer
33 views

If $P(x)$ and $Q(x)$ are both factors of $H(x)$

$P(x)$ and $Q(x)$ are both quadratic polynomials and both are factors of a cubic polynomial $H(x)$ such that: $$H(x) = (x - a)P(x) \space \text{AND} \space H(x) = (x - b)Q(x)$$ For distinct $a,b$ ...
5
votes
7answers
3k views

How do I come up with a function to count a pyramid of apples?

My algebra book has a quick practical example at the beginning of the chapter on polynomials and their functions. Unfortunately it just says "this is why polynomial functions are important" and moves ...
5
votes
1answer
78 views

Polynomial between $0$ and $1$ that produces largest integral

Question: Let $n\in \mathbb{N}$. Find the polynomial $p(x) = \sum_{i=1}^n a_ix^i$ that satisfies $p(1) = 1$ (and $p(0)=0$ since we already have $a_0=0$) $p(x) \in [0,1]$ for all $x\in [...
0
votes
1answer
29 views

What is a formal model for equations in non-commutative ring?

The standard formal modeling for polynomials is the polynomial ring $R[X_1,...,X_n]$ which is a monoid ring $R[\mathbb{N}^n]$ over an rng $R$. Under this construction, it is possible to commute $X_i$'...
1
vote
1answer
54 views

Derivation of Viète's Theorem or Formulas

I had problems finding the proof of an equation in G. Polya "Mathematics and Plausible Reasoning" p. 18 that upon a little bit of research turns out to be Viète's Theorem: Given a polynomial, $$a_0 +...
1
vote
2answers
47 views

$5x\big(1+\frac{1}{x^2 +y^2}\big)=12$ ; $5y\big(1-\frac{1}{x^2 +y^2}\big)=4$ find $x$ and $y$

I already tried to solve using substitution and cross multiplication method . I got the first simplified (1)$$\frac{12}{5x}=1+ \frac{1}{x^2} +y^2$$ (2) $$\frac{4}{5y}=1- \frac{1}{x^2+y^2}$$ Adding (...
-1
votes
1answer
214 views

Radical solution to a polynomial quartic equation

Consider the following quartic equation: $$x^4 + rx^3 + r^2x^2 + r^3x + r^4 - 1 = 0$$ By Lodovico Ferrari solution, this equation must possess four radical solution provided that $r$ is a rational ...