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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How do I fit $f(x) = \exp(a+bx+cx^2 +dx^3)$ to two points? $x, f(x)$ and $f'(x)$ are known.

In the past I've fit polynomials by solving the set of equations. I can fit $f(x) = \exp(a+bx)$ to point $A$ and $B$ where I know $x$ and $f(x)$ for both points. If I want to fit to a specific ...
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Polynomial Interpolation When $y_i$'s are Permuted

Recall, if we have a $d$-degree polynomial $f$, evaluate it at $\textbf{x}=(x_1,\ldots,x_n)$ we would get $\textbf{y}=(y_1,\ldots,y_n)$, where $f(x_i)=y_i$ and $d+1 \leq n$. The reverse is also true, ...
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40 views

Is every algebraic integer a sum of roots of $x^n - a$?

A complex number is said to be an algebraic integer if it is a root of a monic polynomial with integer coefficents. For example any root of the polynomial $x^n - a$ for $a \in \mathbb{Z}$ is an ...
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Proof for two monic polynomial gcds, $d$ and $d_0$, if $d|d_0$ and $d_0|d$, then $d=d_0$

This is an extension to this, that is covered in my higher linear algebra course. I know if $d$ and $d_0$, both $\in \mathbb{F}[x]$ are monic and gcds of some polynomials $g$ and $f$ in ...
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Solve a system of three equations [on hold]

$x=\dfrac{a-.5c+.25d}{a+c+d}$ $y=\dfrac{.5c\sqrt{3}+.25d\sqrt{3}}{a+c+d}$ $z=a+c+d2$ How do I make it so that only $x$, $y$, and $z$ are on the Right-Hand-Side of the equation while only $a$, $c$, ...
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1answer
28 views

Minimize distance between polynomials, of a certain form, with Laguerre polynomials

A typical problem that I may encounter on an upcoming test looks like this: Find the polynomial $P(x)$ of a degree less than or equal to three that minimizes $$\int_0^\infty (x^4 - ...
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24 views

How to find $p\in \Bbb C[X]$ given $p(p(X))$

Assume you're given $p(p(X))$ in the form $$p(p(X))= \sum_{iā‰„0} a_i X^i$$ Is there any quick algorithm to retrieve $p$? What can be said about the degree of $p(p)$ I think it's twice the degree of ...
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1answer
26 views

Polynomials with range containing an arithmetic progression

Can I find a polynomial in a second degree in two variables from the values of which can be found an infinite arithmetic progression? Thank you!
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32 views

Eigenvectors of the companion matrix

Suppose one has an Hermitian square matrix $A$ with $p$ is the characteristic polynomial $$ p(x)= a_0 + a_1 x + \cdots + a_{n-1}x^{n-1} + x^n ~, $$ and define the companion matrix of $p$ as $$ ...
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1answer
49 views

what is f(x) < 0 asking for?

I'm trying to answer a question that says, State where $f(x)<0$ using any correct notation and I do not know what it is asking for. The question provides me a graph going from quadrant 2 to 4, and ...
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20 views

Zero locus of 2-variate real polynomial are smooth curves

This seems like it should be an easy question, and probably already has already had answer in advanced mathematics, but I only know some basic calculus, so I would like to know how do I go about doing ...
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2answers
32 views

$\frac{y-b}{r}=\frac{y}{s}$ to $y$ for finding the closest point on a line, from a point.

$$r=sy^2-sby$$ How do I get $y$ on one side? Originally I had: $\dfrac{y-b}{r}=\dfrac{y}{s}$
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2answers
36 views

Proof that the Runge Phenomenon occurs

Is there such a proof that states that the Runge Phenomena will always occur when interpolating with higher order polynomials or is this just observed empirically?
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1answer
43 views

For which elements $t$ in a finite field $\mathbb{F}_{p^n}$ is $t^2 - 4$ a square?

That is, how to characterize the elements $t \in \mathbb{F}_{p^n}$ for which there exists $x \in \mathbb{F}_{p^n}$ such that $t^2 - 4 = x^2$?
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1answer
44 views

Show that $(t^m-1)/(t^n-1)$ is a square if and only if $(\exists s \in \mathbb{Z})\ m=np^s$

I want to show the following lemma: Assume that the characteristic of the field $F$ is $p$ and $p>2$. Then $(t^m-1)/(t^n-1)$ is a square in $F[t, t^{-1}]$ ($F[t,t^{-1}]$: the polynomials in ...
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Polynomial Interpolation and Data Integrity

This question is about polynomial interpolation and security. Please consider a scenario where we have a polynomial $f$, one of whose roots is $a$. We evaluate it at some ...
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1answer
59 views

Determining polynomial values

The polynomial has been edited to include the "x" term $R(x)= x^4+Ax^3+Bx^2+10x-1$ has a remainder of $-15$ when divided by $x+1$ and a remainder of $39$ when divided by $x-2$. Determine $A$ and ...
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1answer
82 views

Equality regarding Bernstein polynomials

The Bernstein polynomials are defined like this: $b_k(m,x)= {{m}\choose{k}} x^k(1-x)^{m-k}$, if $k<m$ I want to prove that $\sum\limits_{j=k}^m b_j(m,x) = m {{m-1}\choose{k-1} }\int\limits_{0}^x ...
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23 views

How would one define polynomials over the projective line $P_K^1$

May $K$ be a field. If I set $\varXi=(X:Y)$ as a "projective variable" and "projective coefficients" $a_k=(x_k:y_k)\in P_K^1$ - may I then write a polynomial map $P_K^1\longrightarrow P_K^1$ in a form ...
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1answer
57 views

Functional equation: Finding $f(100)$

A polynomial of degree 98 such $f (k)=1/k$ for $k=1,2,3...,98,99$ exists. How to find $f(100)$? What are the possible methods ?
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35 views

Decomposition of a homogeneous polynomial

Let $k$ be a field. Suppose I have a homogeneous polynomial $f$ in $k[x,y,z]$. If $f$ is reducible, does it always decompose as a product of homogeneous polynomials? Thanks!
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1answer
32 views

Does every non trivial variety in $\mathbb{R}^n$ have empty interior?

By this question, we know that a non-trivial affine variety in $\mathbb{C}^n$ has empty interior. But the argument uses the (strong) fact that a holomorphic function vanishing in a non empty set $U$ ...
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1answer
64 views

Factoring $x^{15}āˆ’1$ into irreducible polynomials over $\mathrm{GF}(2)$

Factorize $x^{15}āˆ’1$ into irreducible polynomials over $\mathrm{GF}(2)$ The answer is $$(x+1)(x^2+x+1)(x^4+x+1)(x^4+x^3+1)(x^4+x^3+x^2+x+1)$$ but how would I find this? Please help.
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3answers
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How does determining the area of rectangle relate to binomial multiplication?

So using the strategy to determine the area of the large rectangle I simply did $10\times10, 10\times2, 10\times4, 2\times4$ to get $168\mathrm{cm}$ total. The next question goes on to ask how ...
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1answer
83 views

Number of polynomial factors of $a^n-b^n$?

This is a number theoretical problem that I discovered myself. Let $f(n)$ be the number of factors of $a^n-b^n$ with integer coefficients when its completely factored. For example: $f(1)=1$, because ...
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2answers
29 views

finding factors of a polynomial

In the math problem in the attached image, it explains how to find the factors of a polynomial whereby every possible factor of the function is of the form p/q, where "p is a factor of the constant ...
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38 views

Splitting even degree polynomials

I have an octic equation (degree $8$) and a sextic equation (degree $6$) in $\Bbb Z[x]$ with very large coefficients (size several hundred bits) that I know splits into two quartics and two cubics ...
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66 views

Find all the possible real values for $a,b,c,d$.

Let pairs $(a,c)$ and $(b,d)$ be roots of the equations $x^2 + ax - b = 0$ and $x^2 + cx + d = 0$ respectively. Find all possible real values for $a,b,c,d$.
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1answer
20 views

How to split a quartic into two quadratics?

I have a quartic in $\Bbb Z[x]$ with very large coefficients that I know splits into two quadratics in $\Bbb Z[x]$. What is the best way to do find the quadratics?
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1answer
17 views

Inverting power sum of symmetric polynomial

Suppose I have a set of power sum symmetric polynomial as $$S_p =\sum_i^N x^p_i ~~;~~~~~~~~p=\{1,N\}$$ and I have N of them $\{S_1...S_N\}$ Question is given this, can we find ${x_n=F(\{S_p\})}$? ...
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1answer
99 views

Solve $x^3+x+3=0$ by Galois's theory

Solve with radicals the following equation $x^3+x+3=0$, using Galois Theory. I'm just starting learning this and I do not have many ideas.
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1answer
40 views

polynomial factorization when exponent is not given

How can I factorise this equation, given i already know some of its factors which are: $(a-b)(b-c)(c-a).$ Equation is : $$a^nb^{n-1} + a^{n-1}c^n - a^nc^{n-1} - a^{n-1}b^n - b^{n-1}c^n + b^nc^{n-1}$$ ...
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Matrix polynomial [on hold]

Suppose: ${A_j},{\Delta _j} \in {C^{n \times n}},(j = 0,1,2....m)$ ${\rm{P(}}\lambda {\rm{) = }}{{\rm{A}}_m}{\lambda ^m} + .....{A_1}\lambda + {A_0}$ is a matrix polynomial, and $\lambda $ is a ...
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44 views

Lowest root of a quintic equation with 5 positive roots

I have a quintic equation $$ x^5-a_4 x^4+a_3 x^3-a_2 x^2+a_1 x - a_0=0 $$ with $a_n>0$ real coefficients, and I know that all 5 roots are real and positive (it is a characteristic polynomial). ...
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2answers
61 views

Finding roots of $2x^3-5x^2+18x+45$

solve $2x^3-5x^2+18x+45$ not exactly sure where to start on finding the zeros complex or real. There is one real zero and two complex I know that from graphing just cannot do it on paper to understand ...
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Maple not able to calculate Bernstein polynomial

Hope you can help me on this one. Please look at this simple Maple code: Obviously $B(1)=g(1)=4 \neq 0$. Why is Maple not able to compute this right? Am I doing something wrong? Kind regards PS: ...
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71 views

How to derive the Taylor expansion form of a polynomial expression?

I want to change this polynomial into the form $\sum_{k=0}^m a_k x^k$ $$q_m(x)=\sum_{k=0}^m(-1)^k\binom{2m+1}{2k+1}x^k(1-x)^{m-k}$$ I see no way to do this as I fear one might need intricate binomial ...
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1answer
44 views

Quartic equation or Sextic equation? And how to solve it?

In this arxiv paper (p. 11, eq. (3.2)) the authors claim that equation (3.2) is ... a quartic equation [...] which can be solved explicitly. The equation in question is \begin{equation} ...
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31 views

Describing the graph of a function

For my Algebra II class one of the questions was: Describe the graph of the function $f(x) = x^3 - 18x^2 + 107x-210$. Include the $y$-intercept, $x$-intercepts, and the shape of the graph. And my ...
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1answer
35 views

Prove the theorem of ideal (about g.c.d)

If $p_1,\ldots,p_n$ are polynomials over a field $F$, not all of which are $0$, there is a unique monic polynomial $d$ in $F[x]$ such that (a) $d$ is in the ideal generated by $p_1, \ldots, ...
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1answer
90 views

Find $r$ in the next formula

Lets suppose I have the next values $$0, 7, 8, 5, 6$$ And the next formula $$4250 = \frac{0}{(1+r)} + \frac{7}{(1+r)^2} + \frac{8}{(1+r)^3} + \frac{5}{(1+r)^4} + \frac{6}{(1+r)^5}.$$ What is the ...
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1answer
41 views

How would you divide a polynomial by another polynomial whose power is greater than its nominator? [on hold]

I have a polynomial which is: $$\frac{(x^3-4x)}{(4x^2-4x+1)} = -10$$ Is there a way to do this? I have thought about doing long division which was not helpful...
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1answer
28 views

Functions that are “balanced” on the support of a permutation

Let $F = GF(2^n)$. Let $P(x), Q(x) \in F[x]$ be such that $P(x)$ is a permutation, while $Q(x)$ is not a permutation. For $\lambda \in F^*$ define the function $g_\lambda(x) = Tr(\lambda Q(x))$. Let ...
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Runge Phenomena and Taylor Expansion

From The Weierstrass Approximation Theorem Vs The Runge's Phenomenon: We contrast this to polynomial interpolation: this is a specific method for generating a sequence of polynomials that ...
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1answer
17 views

Notation for polynomials and equating coefficients

I am reading a paper that defines $P_k(s|t)$ as a polynomial of degree $k$ in $s$ given $t$. Does this mean that each term is of the form $f_{k}(t)s^{k}$? (What does "given $t$" mean?) The paper ...
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1answer
34 views

Understanding a simple proof about minimal polynomials

Let $T \colon V\to V $ be a linear operator, where $V$ is a vector space over $F$. Suppose that the minimal polynomial $M(t)$ of $T$ can be factored into the product of two coprime and monic ...
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Does there exist a proof of the Abel-Ruffini theorem without using Galois theory?

Question: Does there exist a proof of the Abel-Ruffini theorem without using Galois theory? Original: Prove that there exists a polynomial $P$ with $\deg P \geq 6$ such that the minima and maxima ...
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83 views

Prove an equality

If $a+b+c=0$ prove that $\frac {(a^4 +b^4 +c^4)}{2}=\frac {(a^2+b^2+c^2)}{2^2}^2$ I have expanded the right side and have got this far: $a^4+b^4+c^4+2(a^2b^2+a^2c^2+b^2c^2)$ I need $a^2=b^2=c^2$ ...
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0answers
40 views

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 ...
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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 ...