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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729 views

Relative Maxima/Minima of polynomial functions

I am taking the Pre Calculus 12 course online. I came across this concept that the online material teaches in 3 different ways, and each one contradicts the other. I find this extremely frustrating. ...
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49 views

Sum of square of absolute values of roots of a polynomial

If $\alpha_1,\dots,\alpha_n$ are roots of a polynomial $$P(z)=z^n+a_1z^{n-1}+\dots+a_{n-1}z+1,$$then how can one express the sum $$|\alpha_1|^2+\dots+|\alpha_n|^2$$in terms of $a_i$'s? Thanks.
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34 views

What is it called when we interpolate a point INTO a grid…

Consider a uniform 2D grid, where each $(x,y)$ value on this grid has a corresponding value. So, if I want to find the value, $v$ (unknown) of a point that exists at some arbitrary co-ordinate $(x,y)$ ...
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5answers
382 views

system of equations

$$u + v = 2$$ $$ux + vy = 1$$ $$ux^2 + vy^2 = - 1$$ $$ux^3 + vy^3 = -5$$ Can anyone give me a hint for solving this ? I'm kinda stuck.
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0answers
45 views

How to prove this polynomial has an imaginary root? [duplicate]

How can we show that the polynomial $a_nx^n + a_{n-1}x^{n-1} + a_3x^3 + x^2 + x + 1 = 0$, where $a_i\in \Bbb R$, $i=3,...,n$ has an imaginary root?
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1answer
25 views

How to construct a polynomial from a radix-term?

A term only composed of the following operatings shall henceforth be called a radix term because I don't know how these terms are called. A radix term $t$ is either an integer or a sum of two radix ...
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28 views

Minimal Polynomial

Determine the minimal polynomial of $\frac{1}{\sqrt[5]{2}}+\frac{1}{11}$ over $\mathbb{Q}$. Put $x=\frac{1}{\sqrt[5]{2}}+\frac{1}{11}$. Put $x=\frac{1}{\sqrt[5]{2}}+\frac{1}{11}$. We need to ...
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1answer
26 views

A conceptual question about inner product spaces for continuous functions and bases.

I hope this makes sense, I had a test yesterday and I couldn't answer this question. The question was laid out something like the following: I was provided a basis for 5th degree polynomials, ...
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1answer
16 views

estimating the roots of $ \epsilon z^n + p(z)$

I have a polynomial $p(z)$ of degree $n-1$ with known roots $z_1, \dots, z_{n-1}$. How I add the monomial term $a z^n$. What are the roots of $$ p_1(z) = p(z) + \epsilon z^n $$ In terms of the ...
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20 views

Cyclotomic polynomial,

Show that $\displaystyle X^n-1=\prod_{d\mid n}\Phi_d(X)$. We have that $$\Phi_n(X)=\prod_{\underset{\gcd(i,n)=1}{1\leq i\leq n}}(X-\zeta_n^i)$$ where $\zeta_n=e^{\frac{2i\pi}{n}}$ therefore, we ...
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27 views

How to find the Coefficient of the Quadratic Term?

Given $4x^3 +bx^2+cx+d$ and two roots of this cubic function $(0,0)$ and $(2,0)$ Find the coefficient of the quadratic term? When I first read this I had no idea how to solve this and still ...
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1answer
20 views

A polynomial can be written as the difference of sub-harmonic functions

Let $\Omega\subset \mathbb R^N$ open bounded be given, I am trying to prove that first any Polynomial can be written as difference of two sub-harmonic functions, and then for any continuous function ...
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1answer
80 views

Why is polynomial long division being taught in schools instead of Horner's method? [closed]

The Horner´s method is by a long shot easier than the Polynomial long division and serves the same purpose. Why isnt it being taught in school (in germany at least)?
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15 views

Is this a valid way to extend the proof of the insolubility of the quintic?

I'm just musing here, this is only (barely even) a half-formed idea, but I'm just wondering if it's at all a valid train of thought. The proof I read of the insolubility of the quintic polynomial ...
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1answer
400 views

Can you help me reverse the Minimum Curvature Method?

The minimum curvature method is used in oil drilling to calculate positional data from directional data. A survey is a reading at a certain depth down the borehole that contains measured depth, ...
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1answer
50 views

Represent $x^n + x^{n−1}y + x^{n−2}y^2 + \dots + xy^{n−1} + y^n$ as $\sum_{i=1}^k g_i(x)h_i(y)$

Problem. Given a natural number $n$, consider the function $$ f_n(x, y) = x^n + x^{n−1}y + x^{n−2}y^2 + \dots + xy^{n−1} + y^n $$ of two real variables. Find the minimal number $k$ for which there ...
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2answers
26 views

What is the minimum degree for a polynomial to pass through points with defined slopes [duplicate]

I'm having some difficulty solving this problem. The information I have is the following: What is the minimum degree for a polynomial for it to pass through points $A(x_1,y_1)$ and $B(x_2,y_2)$ with ...
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5answers
15k views

How to solve an nth degree polynomial equation

The typical approach of solving a quadratic equation is to solve for the roots $$x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$$ Here, the degree of x is given to be 2 However, I was wondering on how to solve ...
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114 views

Finding the roots of $x^n+\frac{1}{x^n}=k$

Find the roots of $$x^n+\frac{1}{x^n}=k$$ when $n$ is an integer number and the $k$ is positive integer number. So far I found one root which is $x=\frac{1+\sqrt{5}}{2}$ when $n$ is even.
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38 views

Roots of a polynomial $f(x)$ in $\mathbb{C}[x]$

"Find all the roots of the polynomial $f(x)=x^2+(3i-2)x-2(1+i)$. Why does the answer not violate the $Conjugate \space Roots \space Theorem \space (CJRT)$" I tried using the quadratic formula and ...
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1answer
77 views

How can I convince students a certain polynomial equation is symmetric?

How can I convince students that $p(x)=0$ is a symmetric equation if they ask me, where $p(x)$ is polynomial of degree $n$ with reals coefficients. For example : $A(x)=2x^4-9x^3+8x^2-9x+2=0 $ is ...
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Ideals, prime ideals and maximal ideals of the ring $K=\mathbb R[x]/\langle (x^2+1)(x-2)^2\rangle$ [closed]

I am trying to find the ideals, prime ideals and maximal ideals of this ring: $K=\mathbb R[x]/\langle (x^2+1)(x-2)^2\rangle$. I am fairly fluent in abstract algebra though ideals are my huge ...
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1answer
31 views

Is an ideal prime when its complex extension is prime?

Let $I = \langle f_1,\dots,f_k\rangle$ be an ideal in $\mathbb R[x_1,\dots,x_n]$. The same $f_i$ generate an ideal $\widetilde I$ in $\mathbb C[x_1,\dots,x_n]$. When $\widetilde I$ is prime in ...
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2answers
23 views

Approximation of roots of a polynomial

Consider the polynomial: $x^7−(3/2)x^6−(43/4)x^5+(115/8)x^4+(135/8)x^3−(61/8)x^2−(81/8)x−(9/4)$ How can I approximate its roots without using Newton's method? (Using Newton's method I got a root of ...
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54 views

how can I find the smallest integer n such that a polynomial divides x^n-1

I have a simple question.. Assume that I have an arbitrary polynomial $f$ in $F_q[x]$. Is there a practical way to find the smallest integer $n$ for which $f$ divides $x^n-1$ ? A small example ...
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5answers
75 views

$x^3-3x^2+4x-2$ cannot be factored over $\mathbb R$

I'm new to the site, and I need a bit of help from you. How can I prove that the polynomial: $f(x)=x^3-3x^2+4x-2$ cannot be factored as a product of polynomials of degree 1 with real coefficients? ...
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25 views

Polynomial growth, using the Cauchy Integral Formula,

Is this a true statement in Complex Analysis? If a function grows like a polynomial, then it is a polynomial. Or, is it really: if a function grows like a polynomial at infinity, then it is a ...
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1answer
42 views

Solving Polynomial in $\sin$ and $\cos$

Given the following equation: $$A \sin(t) + B\cos(t) =C\sin(t)\cos(t).$$ Can we solve the above equation without the need of using general solution of 4-order polynomial equation. We can use ...
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1answer
13 views

To find the number of terms in a polynomial series product?

The question asks me to find the number of terms in the polynomial product expansion : $$ (1 + x^{-1})(1 + x^{-2})(1 + x^{-4})(1 + x^{-8})......(1 + x^{-2^{n}}) $$ I tried multiplying by ...
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51 views

On the location of the roots of a polynomial

Consider the following two polynomials \begin{align} p(s)&:=s^n+\alpha_{n-1}s^{n-1}+\cdots+\alpha_1s+\alpha_0,\\ q(s)&:=s^{n-1}+\alpha_{n-1}s^{n-2}+\cdots+\alpha_2 s+\alpha_1, \end{align} ...
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39 views

Proving a polynomial is prime in $ \mathbb{R}[t]$?

I had gotten this question on a recent assignment, and am confused on how to approach it. Would I need to use Gauss's Lemma? Prove that $t^2 + 1$ is a prime in $ \mathbb{R}[t]$. Prove that no ...
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287 views

Krylov-like method for solving systems of polynomials?

To iteratively solve large linear systems, many current state-of-the-art methods work by finding approximate solutions in successively larger (Krylov) subspaces. Are there similar iterative methods ...
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What can one conclude for a poynomial with the property that p(x)=p(ix)?

From a theorem in the theory of cyclotomic polynomials I deduced that a special polynomial $p(x)$ of even degree $n$ has the property $$p(x)=p(ix)$$ with $i$ being the complex unit. What can one ...
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40 views

how to generate rook polynomial

I've encountered rook polynomials. I just can't seem to understand how to generate them by hand for small examples such as 3x3 boards. Take for instance: $$\begin{matrix} 1 & 1 & 0 \\ 1 ...
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3answers
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What is a primitive polynomial?

What is a primitive polynomial? I was looking into some random number generation algorithms and 'primitive polynomial' came up a sufficient number of times that I decided to look into it in more ...
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Algebraic division with two variables? $\frac{a^3 + b^3}{a+b}$

I know there's a formula for this, but I would like to know how to do algebraic division the long way - would appreciate if you can guide me along. How can I use long division for $$\frac{a^3 + ...
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Degrees of spaces of polynomials

Let $I$ be an ideal in $K[x_1,\dots,x_n]$ where $K$ is a char $0$ field. Let $Z(I)$ be a set of discrete points whose cardinality is exponential in $n$ and spanning $n$ dimensions. Let $P$ be the ...
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1answer
40 views

A sufficient condition to ensure a polynomial to be zero

Let $p_i(x)$, $p(x)$ be real coefficient polynomials. Suppose that $$\sum_{i=0}^{n-1}x^ip_i(x^{in})=p(x^n), (x-1)\mid p(x).$$ Show that $p_i(x)=0$, $1\leq i\leq n-1$. I could only show that ...
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46 views

How do you 'rotate' a polynomial?

I have a polynomial equation: $$y=(-5 \times 10^{-6} \times x^3)+(0.0004 \times x^2)+(0.0582 \times x)-0.4397$$ Is it possible to "rotate" this polynomial curve (maintaining the shape) around the ...
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2answers
58 views

How do I find out if a polynomial is irreducible?

I have this polynomial: $f(x)=x^4+x^3-4x^2-5x-5$. How can I find out if this polynomial is irreducible over the field $Q$ of rational numbers? I know about mod p irreducibility test but it fails in ...
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Prove coefficients of polynomial are elementary symmetric polynomials

I want to show that for the $k$-th elementary symmetric polynomial $s_k:=\sum_{i_1\lt\cdots\lt i_k}X_{i_1}\cdots X_{i_k}\in R[X_1,\ldots,X_n]$ a monic polynomial that factors $\prod_{i=1}^n ...
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1answer
30 views

How to change the gradient of polynomial?

I have a polynomial equation: $$y=(-2 \times 10^{-10} \times x^5)+(1 \times 10^{-7} \times x^4)-(2 \times 10^{-5} \times x^3)+(0.0018 \times x^2)-(0.0156 \times x)-0.164$$ I want to be able to ...
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1answer
55 views

Factor of determinant with identical row

How the following fact applies to determinants (I came across it while solving problems): Consider A is a nxn matrix, the elements of which are real (or complex) polynomials in x. If r rows of the ...
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2answers
25 views

Can all cubic/quartic polynomials be expressed in a form with only one x term?

Quadratic expressions $ax^2+bx+c$ can all be expressed in a form with only one x term: $$a(x+\frac{b}{2a})^2+c-\frac {b^2}{4a}$$ Is the same true for all cubic or quartic expressions? Is there a ...
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Finding Basis for a Radical of an Ideal

I am to find a basis of the following ideal: $$\sqrt{<x^5-2x^4+2x^2-x, \quad x^5-x^4-2x^3+2x^2+x-1>}$$ Truth be told, I'm not entirely confident of my solution. I will present it and then ask ...
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System of 4 quartic equations

\begin{align*}a &=\sqrt{4+\sqrt{5+a}},\\ b &=\sqrt{4-\sqrt{5+b}},\\ c &=\sqrt{4+\sqrt{5-c}},\\ d &=\sqrt{4-\sqrt{5-d}}.\end{align*} Compute $abcd$. I set up each as a quartic and got ...
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1answer
14 views

separable polynomial

How to show that if $K$ is a field of characteristic $p$ with $p$ prime and if $f(X)\in K[X]$ is an irreducible and inseparable polynomials, therefore there exist a $d\in\mathbb N, d>0$ such ...
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2answers
40 views

Prove that for $f,g\in F[x]$, where $F$ is an infinite field, if $f(a)=g(a)$ for infinitely many elements $a\in F$, then $f=g$

Prove that for $f,g\in F[x]$, where $F$ is an infinite field, if $f(a)=g(a)$ for infinitely many elements $a\in F$, then $f=g$. I'm not entirely sure how to tackle the "infinitely many elements ...
5
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1answer
330 views

Irreducible factors of $X^p-1$ in $(\mathbb{Z}/q \mathbb{Z})[X]$

Is it possible to determine how many irreducible factors $X^p-1$ in the polynomial ring $(\mathbb{Z}/q \mathbb{Z})[X]$ has and maybe even the degrees of the irreducible factors? ($p,q$ are a primes ...
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1answer
29 views

How many polynomials in $Z_{p}[x]$ have degree n or less?

For your reference, $Z_{p}[x]$ refers to the set of all polynomials with coefficients integer mod p. To me it seems like this and the degree (power) of the two polynomials are unrelated. What ...