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Questions tagged [polynomials]

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 and checking for irreducibility.

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find the answer in terms of a and b only (a and b are roots of $\ x^4 + x^3 - 1 = 0$

if a and b are the two solutions of $\ x^4 + x^3 - 1 = 0$ the solution of $\ x^6 + x^4 + x^3 - x^2 - 1 = 0$ is?? well im not able to eliminate or convert $\ x^6$ please help
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3answers
28 views

Proof request: If a function satisfies a polynomial recurrence then it is a polynomial

Suppose we would like to find a formula for some interesting real-valued function on the positive integers $f(n):\mathbb{Z}^+\rightarrow\mathbb{R}$, and we have the inductive equation $$\forall n\in\...
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4answers
32 views

Why $(x^5+x)^{\frac{1}{3}}=x^\frac{5}{3}(1+\frac{1}{x^4})^{\frac{1}{3}}$?

My teacher today wrote the following equation: $(x^5+x)^{\frac{1}{3}}=x^\frac{5}{3}(1+\frac{1}{x^4})^{\frac{1}{3}}$. Why is this true?
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0answers
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computing polynomials whose roots are the vertices of 4-polytopes of circumradius one interpreted as quaternions

Suppose we have the vertices of 4-polytopes which have been scaled so that their are on the unit sphere. Interpret them as quaternions. If these have been computed before, I'm looking for a ...
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1answer
62 views

Let $f(x)$ be a $n$ degree polynomial function having $n$ real and distinct roots. If $g(x) = f'(x) + 100f(x)$

Let $f(x)$ be a $n$ degree polynomial function having $n$ real and distinct roots. If $g(x) = f'(x) + 100f(x)$, then minimum number of roots that $g(x)$ must possess is: $\text {a) n}$ $\...
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1answer
25 views

Proof that e is transcendental in Herstein's Topics in Algebra (1st ed)

In the proof of Theorem 5.F. page 177. page 176 page 177 From the constructed $F(x)$ from $f(x),$ how can we choose $$f(x)=\frac{1}{(p-1)!}x^{p-1}(1-x)^{p}(2-x)^{p}\cdots(n-x)^{p}$$ where $p>n$ ...
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1answer
83 views

When is $x^3+x+n$ reducible?

I am trying to work out when $f=x^3+x+n$ is reducible over $\mathbb{Q}$, with $n\in \mathbb{Z}$. So far, I have reduced to irreducibility over $\mathbb{Z}$ by Gauss's lemma, and have that $f=(x+a)(x^2-...
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1answer
35 views

Nature of roots [on hold]

If ${{A_1},{A_2},{A_3},....,{A_n},{a_1},{a_2},{a_3},....,{a_n},a,b,c \in R}$ , show that the roots of the equation ${\frac{{{A_1}^2}}{{x - {a_1}}} + \frac{{{A_2}^2}}{{x - {a_2}}} + \frac{{A_3^2}}{{x - ...
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0answers
19 views

Showing $x$ and $y$ are permutations of one another for Cauchy joint distribution

This is similar to a question I just asked but the method there does not work here. Consider the joint distribution of iid Cauchy random variables $$f(x|\theta) = \prod_{i=1}^n \frac{1}{1+(x_i-\...
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Proof check: showing $x$ and $y$ must be permutations of one another given…

In what follows $x_i, y_i, z> 0$. I want to show that if $$\prod_i \frac{1-x_iz}{1-y_iz} = C(x,y)$$ where $C(x,y)$ is a constant independent of $z$, then $x=(x_1,\dots,x_n)$ and $y=(y_1,\dots,...
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3answers
41 views

Polynomial division problem- find the degree of the remainder

Let $r(x)$ be the remainder when the polynomial $x^{135}+x^{125}-x^{115}+x^5+1$ is divided by $x^3-x$. Then a. $r(x)$ is the zero polynomial b. $r(x)$ is a nonzero constant c. the degree of $r(x)$ is ...
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0answers
79 views

Number of real zeroes of iterated polynomial: $x^3-2x+1$

If $P(x)=x^3-2x+1$, define $z_n$ as the number of real roots of the polynomial $P^{\circ n}(x)$, where the superscript denotes $n$-fold composition. Can we find a general formula for $z_n$, or perhaps ...
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1answer
47 views

Is there a way to simplify $(x+y)^z$ using only basic functions? [on hold]

So no Big-O notation, Re(), derivatives, only stuff you could find a scientific calculator.
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0answers
43 views

A question about how to show two polynomials are not equal (follow-up)

This is a follow-up of my previous question, but it is self-contained. Let $f\in k[x_i,i=0,\ldots ,n]$ be a a polynomial which contains $x_0$ (i.e. polynomial like $f=x_1$ is not allowed), where $k$ ...
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2answers
46 views

A question about how to show two polynomials are not equal

Let $f\in k[x_i,i=0,\ldots ,n]$ be a a polynomial which contains $x_0$ (i.e. polynomial like $f=x_1$ is not allowed), where $k$ is a field with characteristic $0$. We define a map $$Q_a: k[x_i,i=0,\...
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1answer
17 views

Given that f(x) is a quadratic function and that f(x) is only positive when x lies between -1 and 3, find f(x) if f(-2) = -10.

What i did: $f(x)=ax^2+bx+c$ $f(-2)=4a-2b+c=-10$ $f(0) =c > 0$ $f(1) =a+b+c > 0$ $f(2) =4a+2b+c > 0$ I thought using $b^2-4ac = 0$ for $f(-2)$ but its wrong since I am getting c = -...
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1answer
24 views

Two quadratic functions with simillar coefficients and a square of their 'b' coefficient

I have quite an interesting problem to share today. I have noticed, that when I pick two quadratic equations, say: $x^2-mx-n=0$ and $x^2-mx+n=0$ Where $m, n$ are natural numbers, and I assume both ...
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1answer
25 views

Continuity of supremum of polynomial

Prove: Let $A \subseteq \mathbb{R}$ be a compact set. Prove that the function $f \colon\mathbb{R^{n+1}} \to \mathbb{R}$ $\qquad f(x_0,..., x_n) = \sup_{x\in A} \prod_{j=0}^{n} (x-x_j)$ is continuous....
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1answer
37 views

The number of attracting fixpoints?

Let $f(z)$ be an entire function of order $0$ with at least $3$ zero’s. $$ f(z) = a_0(1 - a_1 z)(1 - a_2 z)(1 - a_3 z)... $$ There are 2 cases : A) $f(z)$ is polynomial of degree $3$ or higher. B) $...
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2answers
39 views

Can Abel-Ruffini be “seen” by looking at the graphs of polynomials?

The Abel-Ruffini theorem states that there is no algebraic solution – that is, solution in radicals – to the general polynomial equations of degree five or higher with arbitrary ...
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1answer
1k views

A visually guided proof of the fundamental theorem of algebra?

A complex root of a polynomial $P(z)$ is a pair of real numbers $u,v$ that simultaneously make the real part and the imaginary part of $P(z)$ zero. The zeros of the real part and the imaginary part ...
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2answers
48 views

Prove or disprove inequality with 3 variables

I was trying to solve the inequality $$a-\sqrt[3]{a^3-c\cdot a^2}<b-\sqrt[3]{b^3-c\cdot b^2}$$ where $a>b>0$ and $c>0$. I managed to pack the part inside the cube root: $$a-\sqrt[3]{a^2(a-...
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0answers
11 views

Finding the equation of a quartic given its turning points

I'm trying to produce some generalised code to find the equation of a polynomial, given its turning points (and order). I know that for a cubic this can be done using Gaussian Elimination to solve a ...
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0answers
14 views

Advanced Functions Determine the value of m in two polynomials.

The remainder when $f(x) =x^2+mx+1$ is divided by $x-1$ is equal to the remainder when $g(x)=mx^2+x+3$ is divided by $x-2$. Determine the value of $m$.
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1answer
15 views

Intersection of polynomials with positive coefficients and different degrees

This is a variation of this question, but in my case I want to know if two polynomials of different degrees and non-negative coefficients can have more than one intersection in the positive $x$-axis. ...
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0answers
74 views

The function $G:\mathbb{Q}^n \rightarrow \mathsf{FinGroup}$

Consider the function $G:\mathbb{Q}^{n} \rightarrow \mathsf{FinGroup}$ which sends $\langle a_0,a_1,\dots,a_{n-1}\rangle$ via the polynomial $P(x) = a_{n-1}x^{n-1} + \dots + a_1x + a_0$ to its Galois ...
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0answers
33 views

Is there a notion like “graded linear independence”?

I am looking for a suitable notion to discribe the following property for $\{f_i\}$. Let $R=k[x_1,\ldots,x_n]$, $\{f_i\}$ is a set a some homogenous polynomials of degree $2$ (or more general, of ...
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1answer
41 views

Symmetries of polynomials

Polynomials over $\mathbb{Q}$ can exhibit "geometrical symmetries" in different ways, graph-wise (as functions $P:\mathbb{R} \rightarrow \mathbb{R}$) and root-wise: $P(x) = P (-x)$(axial symmetry ...
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0answers
30 views

Calculating Hermite Expansion Coefficents of $|x|$

I'm struggling to calculate the coefficents for the Hermite Expansion of the absolute value function and the indicator function $x \mapsto \mathbb{1}_{|x-u|\leq \delta}$ Background: I know, that for ...
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0answers
31 views

General findings about Galois groups

To get a better feeling for Galois groups I'd like to know some general cases that allow to tell a Galois group from a polynomial and vice versa. The most simple example I came about is $P(x)\in \...
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2answers
49 views

Is this a correct interpretation of the fundamental theorem of algebra?

I tried reading the Wikipedia page but it's stated in terms of complex roots, and I don't really understand how that relates to the following proposition: if a real valued polynomial: $$\sum_{i=0}^n ...
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1answer
58 views

is it true that $P(x)=x-3$ the only solution to this problem?

Question: Find all monic polynomial $P(x)$ with integer coefficients such that there exists a natural number $L$, satisfying: $$p\ | \ 2\times (P(p)!)+1$$ for every prime number $p$ greater than $L$ ...
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1answer
28 views

Why this $12b^2<49ac< \frac{49}{4}b^2$ inequality given ? What is the use of it?

Question : Let $\alpha$ and $\beta$ be roots of $ax^2+bx+c=0$. Show that $(4\alpha-3\beta)(4\beta-3\alpha)=\frac{49ac-12b^2}{a^2}$. If $12b^2<49ac< \frac{49}{4}b^2$ , then show that $\...
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2answers
31 views

Are Laurent series called polynomials?

Traditionally, polynomials cannot have negative exponents. So what gives? Inspired by this.
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0answers
13 views

Zeros of orthogonal polynomial family

Let {$\phi_n(x)$} be an orthogonal family of polynomials for a weight function $w(x)$ on [a, b]. Denote the zeros of $\phi_n(x)$ as $$ a < z_{1,n} < z_{2,n} < ... < z_{n,n} < b$$ Show ...
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0answers
14 views

Extended GCD of two zero polynomials over finite field

Extended GCD of two polynomials $a$ and $b$ results in two polynomials $s$ and $t$ so that $as + bt = \text{gcd}(a, b)$. What convention makes most sense when both $a$ and $b$ are zero? I found that ...
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1answer
40 views

Proving $\max\{|P(x)|,1\leq x\leq2\}\leq C_N·\max\{|P(x)|,0\leq x \leq 1\}, P\in E_N$

Let $E_N$ be the space of polynomials $P(x)$ of order $\leq N$. Prove there exist a constant $C_N>0$ so that $$\max\{|P(x)|,1\leq x\leq2\}\leq C_N·\max\{|P(x)|,0\leq x \leq 1\}, P\in E_N$$ What I'...
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0answers
19 views

Sturm sequence computation using Chebyshev polynomials

According to this paper by B. Gleyse a (partial) Sturm sequence of a polynomial $p_0\in\mathbb{R}[x]$ of degree $n\in\mathbb{N}$ can be computed in the following way: Find coefficients $a_0,\ldots,...
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1answer
28 views

Methods to compute a Sturm sequence

Are there any known methods of computing a Sturm sequence of a polynomial $p$ other than the standard algorithm of applying Euclid's algorithm to $p$ and $p'$? I am asking this question because I ...
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0answers
29 views

Interpolating a number of equidistant points.

I have the coordinates: $(0,3125),(1,3125), (2,2500), (3,1500), (4,600), (5,120), (6,0), (7,0), (8,0), (9,0), ...$ And I want a way to construct a smooth curve through the points that increases in ...
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1answer
20 views

Let $F(x)=x^4-bx^3-11x^2+4(b+1)x+a$.Find $a,b$

Let $F(x)=x^4-bx^3-11x^2+4(b+1)x+a$. Find $a,b$ It’s given that $F(x)$ is a complete square of a quadratic polynomial and $(x+2) $ is a factor of $F(x)$ My attempt : I can write $F(x) = (x+2)^2(Ax+...
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1answer
22 views

Find Divisible polynomial [on hold]

For $P(x) = x^3+ax+b$ and $Q(x) = x^2-3x+2$. Find $a$, $b$ such that with all number $x$ then $P(x) \,\vdots\, Q(x)$.
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1answer
26 views

Is there a relationship between the eigenvalues of convex combinations of two matrices and roots of convex combination of their polynomials?

Let $A_1, A_2 \in M_n(\mathbb R)$ and denote $A := (1-\lambda)A_1 + \lambda A_2$ where $\lambda \in (0,1)$. Let $p_1(t) = \chi_{A_1}(t)$ and $p_2(t) = \chi_{A_2}(t)$ be the characteristic polynomials. ...
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2answers
48 views

Does $f(x)=x^3+x^2-8x+1$ have rational roots?

Problem Does $f(x)=x^3+x^2-8x+1$ have rational roots ? Attempt to solve A citation from our lecturer Possible rational roots are in form of: $$ \frac{\text{factor of constant}}{\text{factor of ...
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0answers
8 views

Polynomial function with no solution using Ruffini's rule [on hold]

im trying to solve algebraically the following polynomial by using Ruffini's rule: $2x^3- \frac {x^2}{2}-12x+1=0$. Unfortunately i can't find any way to solve this without involving calculus and ...
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1answer
32 views

Steps towards the Galois group of permutations of the roots of a polynomial

I'm having a hard time to understand Galois theory and its motivation from a beginner's perspective. (1) What I take for granted is the fundamental theorem of algebra, thus a polynomial of degree $n$ ...
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3answers
27 views

Isomorphism between $K[X]/P(X)$ and $K(A)$ for $A$ a root of the irreducible polynomial $P(X)$ over $K$

I'm afraid the answer to this question should be clear to anyone who knows a little bit of algebra, field theory, field extensions, and polynomials. In his answer to a question about Galois theory ...
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2answers
36 views

What can be said about the roots of $acx^4 + b(a + c)x^3 + (a^2 + b^2 + c^2 )x^2 + b(a + c)x + ac$?

Let a, b and c be real numbers. Then the fourth degree polynomial in $x$, $acx^4 + b(a + c)x^3 + (a^2 + b^2 + c^2 )x^2 + b(a + c)x + ac$ (a) Has four complex (non-real) roots (b) Has ...
0
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1answer
28 views

Need a quadratic function that is larger than zero when variables are unequal

I have a series of variables $a$, $b$, $c$, $d$, .... that are all positive and between zero and one, i.e., $0 \le a \le 1$, etc. I need a quadratic function that is larger than zero if there is any ...
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0answers
25 views

Can an algebraic function be written as a composition of polynomials and their inverses?

For a two variable polynomial $f(x,y)$ which is not constant with respect to $y$, we have an algebraic function $g(x)$ such that for some open subset of $\mathbb{C}$, $f(x,g(x)) = 0$. My question is ...