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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Please help solve this equation [on hold]

$$3.8r - 0.057r^2 + 0.00038r^3 + 0.00000095r^4 = 95$$ Please help.
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ideals of polynomial ring with complex number coefficients

Let $\mathbb{C}[x,y]$ be the polynomial ring with variables $x,y$ and coefficient in $\mathbb{C}$. Let $f,g\in \mathbb{C}[x,y]$. Let $(f,g)$ be the ideal of $\mathbb{C}[x,y]$ generated by $f,g$. ...
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What does a polynomial look like under projection of underlying space?

Consider a multivariate polynomial in $F:\Bbb R^3\rightarrow\Bbb R$, $F\in\Bbb R[x,y,z]$ with prescribed values over a sphere in $\Bbb R^3$. Consider standard Riemann projection from $\Bbb ...
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1answer
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Find the remainder when $x^{100} + 2x + 10$ is divided by $x − 11$ in $\mathbb Z_{17}[x]$

Find the remainder when $x^{100} + 2x + 10$ is divided by $x − 11$ in $\mathbb Z_{17}[x]$ I simplified $x^{100} + 2x + 10$ to $x^{15} + 2x + 10$ and $x − 11$ to $x+6$ to be in $\mathbb Z_{17}$. ...
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1answer
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Ring polynomial kernel generators

This is the textbook question: Q: Find generators for the kernels of the following maps: $\mathbb{R}[x,y] \to \mathbb{R}$ defined by $f(x,y) \mapsto f(0,0)$ $\mathbb{R}[x] \to \mathbb{C}$ defined ...
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About polynomials. If there is not two plynomials with the same grad in S, then S is linearly independent.

The problem states as follows. Let S be a set of polynomials non zero over a field F. If there is not two plynomials with the same grad in S, then S is linearly independent. I tried the following. ...
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2answers
53 views

Is it possible to find the value of $x$ where $e^x$ exceeds $x^{10}$ by hand?

All I managed is to "simplify" the equation $e^x=x^{10}$ to $\frac{x}{\ln{x}}=10$. Is there some way or trick to make the equation look like $x=\dots$? (Solve the equation, in other words.)
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Polynomial and a field [on hold]

How to prove that if a polynomial $$f(x) = ax^3+bx^2 +cx +d,$$ where $a,b,c,d \in K$, where $K$ is a subfield of $\mathbb{C}$, has a root in $K(\alpha)$ then $f$ has a root in $K$. $\alpha \in ...
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1answer
21 views

Solving an equation with complex numbers

I want to use complex numbers to solve the following problem: $x^2 = 95 - 168i$. I am sure there are a few ways of doing this but the way I want to do it is to let $x = a + bi$ and then solve for $a$ ...
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1answer
71 views

Solving the functional equation $f(x^2+f(y))=(f(x))^2+y$

Find all $ f : R\rightarrow R $ such that $f(x^2+f(y))=(f(x))^2+y, \forall\text{ x,y}\in R$ Thanks in advance!
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Discriminant of a polynomial modulo a prime

If $p$ is a prime and divides the discriminant of an irreducible polynomial $f(x)=x^{n}+a_{n-1}x^{n-1}+\cdots+a_1x+a_0\in \mathbb{Z}[x]$ why is then $disc(f(x)\bmod p)=0$? I know that the ...
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3answers
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Symmetric functions and roots of polynomials

$f = x^3-\frac{1}{2}x^2+1$ and their roots $a,b,c$. I want to find polynomial of degree 3 with roots $a^4,b^4,c^4$. I know that i need express $e_i(a^4,b^4,c^4)$ in terms of $e_i(a,b,c)$ those ...
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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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3answers
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Find the least degree Polynomial whose one of the roots is $ \cos(12^{\circ})$

Find the least degree Polynomial with Integer Coefficients whose one of the roots is $ \cos(12^{\circ})$ My Try: we know that $$\cos(5x)=\cos^5x-10\cos^3x\sin^2x+5\cos x\sin^4x$$ Putting ...
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1answer
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Finding roots in finite fields.

On pg. 587 (in the finite fields chapter) of Abstract Algebra, 3rd ed. by Dummit and Foote, the following statement is made: 'If $f_1(x)=x^4+x^3+1$, $f_2(x)=x^4+x+1$ are two of the irreducible ...
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Historical enquiry about polynomials

I'm researching the historical use of geometry to find solutions to polynomial equations. I'd like to ask for those familiar with this topic, how would you describe the use of geometry to solve ...
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1answer
53 views

How can I maintain notes while self studying Maths?

Thank you for stopping by this thread. I'm an engineering student rekindling an interest in Maths. I just love studying Maths in my free time (and sometimes it trespasses into my non free time). I ...
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3answers
38 views

When can I divide both sides of an equation if one side is zero

Where K is some positive Integer For the following examples: $$ K(a+b)(p+q)=0 $$ $$ Ka^2+Kbx+Kc=0 $$ Can I just divide both sides of the equation by K (dividing into 0 on the right) and effectively ...
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1answer
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Show that $B^{(1)} = 0$ and $B^{(2)}$ have basis $\{[x_i,x_j]; i>j\}$

Definition: A polynomial $f \in K \langle X \rangle$ is called a proper polynomial, if it is a linear combination of products of commutators: $$f(x_1,x_2,...x_n) = \sum ...
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3answers
42 views

$\mathbb Q [\sqrt{2} i]$ contains neither $\sqrt[4]{2}$ nor $\sqrt{2}$

I want to prove that $x^4-2$ is irreducible over $\mathbb Q [\sqrt{2} i]$. In order to verify it has no linear factors and quadratic factors, I need to show $\mathbb Q [\sqrt{2} i]$ contains neither ...
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1answer
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can I draw a 2nd degree 2 dimensional surface trough 3x3 points, like I can draw a 2nd degree polynomial through any 3 points

A Nth degree polynomial f(x) fitting N+1 points, say at regular distances like x = 1,2,3,4,5,... can be used conveniently to interpolate for values of x in between the given ones. I have a set of ...
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closed-form expression for roots of a polynomial

It is often said colloquially that the roots of a general polynomial of degree $5$ or higher have "no closed-form formula," but the Abel-Ruffini theorem only proves nonexistence of algebraic ...
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Integral Inequality $\leq n^{3/2}\pi$

$ p(x)\in\mathbb{R[X]} $ is a polynomial of degree $n$ with no real roots. Show that: $ \int\limits_{-\infty}^{+\infty}\dfrac{(p'(x))^2}{(p'(x))^2+(p(x))^2}\,dx \leq n^{3/2}\pi.$ It's easy to see ...
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1answer
856 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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1answer
29 views

How do I prove that $X^{p^n}-X$ is the product of all monic irreducible polynomials of degree dividing $n$?

How do I prove that $X^{p^n}-X$ is the product of all monic irreducible polynomials in $\mathbb Z_p[X]$ of degree dividing $n$? Let $\bar Z_p$ be an algebraic closure of $Z_p$. Define $F=\{x\in ...
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0answers
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How to solve polynomial-exponential equation

I'm trying to solve equations like the following one: $$5 + 3x - 4x^3 = e^{x^2}$$ I've tried using the Lambert W function, but I didn't get any success. I must admit I'm relatively new to Lambert W ...
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1answer
26 views

For a field $K$ is $K\subset{K[X_{1},…X_{n}]}$

Let $K$ be any field and $K[X_1,...X_n]$ the ring of polynomials in $X_1,...X_n$ with coefficients in $K$. I am wondering if $K$ is a subset of $K[X_1,...X_n]$. I believe $K\subset{K[X_1]}$ since ...
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Understanding Primitive Polynomials in GF(2)?

This is an entire field over my head right now, but my research into LFSRs has brought me here. It's my understanding that a primitive polynomial in $GF(2)$ of degree $n$ indicates which taps will ...
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1answer
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Polynomial Equation Solution

Use Demoivre's theorem to show: $cos 7θ = 64 cos7 θ − 112 cos5 θ + 56 cos3 θ − 7 cos θ$ Hence,solve: $128x^7 −224x^5 +112x^3 −14x+1=0$ I've shown the first part and multiplied the equation by 2 and ...
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1answer
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Short method to prove the irreducibility of $x^7+21x^5+35x^2+34x-8$ over $\Bbb Q$

I am given a task to prove that polynomial $f=x^7+21x^5+35x^2+34x-8$ is irreducible over $\Bbb Q$. In my algebra course we learnt reduction and Eisenstein criterion. Eisenstein doesn't seem to work ...
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1answer
30 views

Effect on existing roots of polynomial when adding small higher-order term

How do existing roots of a polynomial change when adding higher-order term with a small coefficient? Given a sufficiently small coefficient of the new higher-order term, the existing roots shouldn't ...
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2answers
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Find Log equation from data points

I have the following data points, (left hand column goes from 0-127, right hand column goes from 30-22000 hz. Is there any calculator I can use to find a "log" function of this data, so that it comes ...
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1answer
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Irreducibility of polynomials of a certain kind

Let us look at factorization over the integers of polynomials of the form $x^n+n$. For the first few values of $n$ we get $x+1$ - irreducible $x^2+2$ - irreducible $x^3+3$ - irreducible $x^4+4$ - ...
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1answer
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What sequence of polynomials is equal to $2^n$ for integers $1$ to $k$?

I am trying to prove to someone that no matter how many terms you have of a sequence you can never be 100% sure of the underlying formula. Consider this sequence: $$2^n=1,2,4,8,16,...$$ But just given ...
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1answer
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Solving the following quartic equation : $x^4- {31\over 4}x^3 + {21\over 4}x + {9\over 2} = 0$

So, I don't know if it's normal but this quartic seems long to solve... When I use the ferrari method, I get the following reduced quartic : $y^4 - {2883\over 128}y^2-{27103\over ...
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4answers
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Is it true that a 3rd order polynomial must have at least one real root?

While solving a problem a friend said - this polynomial is $3^{rd}$ order ($ax^3+bx^2+cx+d$), so it must have a real root. I didn't want to sound stupid and I said sure. I can't figure out if he's ...
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1answer
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$f(x) = k^n$ for infinitely many integers $k$

Let $f(x)$ be a polynomial of $n^{th}$ degree with integer coefficients and let the leading coefficient be 1. Is it true that $f(x) = k^n$ for infinitely many integers $k$ and $x$ if and only if all ...
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1answer
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Find the minimum polynomial of a sum of roots of unity.

Let $ \omega $ be an 11-th primitive root of 1 over $ \Bbb Q $ Let $ \beta = \omega + \omega^9 $ Find $ [ \Bbb Q ( \beta) : \Bbb Q ) ] $ and Find the minimum polynomail of $\beta$. I asked a ...
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Polynomial: Is there a theorem that can save my proof when $K$ doesn't include $\mathbb C$

Suppose $f(x),g(x)\in K[x]$ ($K$ a number field), let $f(x)=x^{3m}+x^{3n+1}+x^{3p+2}$, where $m,n,p\in\mathbb N$, and let $g(x)=x^2+x+1$, prove: $$g(x)\mid f(x)$$ I think this problem is not ...
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Bounds of Zeros of a Higher Degree Polynomial Equation

The Equation is follows: $$\lambda^{k+1}+\frac{A}{A+B} \lambda^{k-l}+\frac{B}{A+B}=0$$ Where $l$ and $k$ are positive integers such that $l<k$. Also A and B are complex numbers. I would like to ...
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Interval of Polynomial Root Finding

Let's say we have a polynomial of a given degree. You don't have any tools to figure out the amount of roots in this polynomial. All you know is the function and you cannot graph it. How would you ...
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1answer
25 views

Interval in which roots lie.

For a quadratic equation, we have several conditions from which we can determine the interval in which the roots lie. eg: If exactly one of the roots of a general quadratic equation lies in the ...
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5answers
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$x^4 + 1$ reducible over $\mathbb{R}$… is this possible?

I am seeing this on a homework and am wondering if this is a typo. I am aware that $x^4 + 1$ is irreducible over $\mathbb{Q}$. I know the following: A polynomial being irreducible over some ring ...
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2answers
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Basis for the vector space P2

I am trying to wrap my head around vector spaces of polynomials in P2. If I represent the polynomial $ ax^2 + bx + c $ with the matrix $ A = \begin{bmatrix} 1,0,0 \\ 0,1,0 \\ 0,0,1 \\ \end{bmatrix} ...
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1answer
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Prove or disprove this relation between one root of the quadratic and the cubic equation of a certain form, and linear recurrences.

It is well known that the n-anacci (higher degree Fibonacci, that is Tribonacci and so on) numbers can be computed in closed form from roots of polynomials in the way Eric Weisstein at Mathworld ...
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1answer
25 views

scaling up data series based on polynomial equation

I have yearly time series data starting from 1990 to 2100 AD (x-axis). The value for 1990 is 0 and 2100 is 700, and it's increasing in each year (but not linearly). Based on this series, I come up ...
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1answer
190 views

“Convex” polynomials

Let me define "convex" polynomials, as the smallest class $\mathcal{C}$ of functions $p:\mathbb{R}\rightarrow \mathbb{R}$ defined (inductively) as: UPDATED (case 0 was missing): 0) $p(x)=x$, i.e., ...
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Given $n \in \Bbb Z$, determine $\gcd(3n^2 + 7n + 4, n + 2)$.

I factored $3n^2+7n+4$ to $(3n+4)(n+1)$ and because there isn't a common factor of those and $n+2$ I said that the gcd is $1$, but is there any othere way to go about it that would come up with a gcd ...
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1answer
81 views

A polynomial that annihilates two other

While studying, I found the following problem: Let $f, g \in F[t]$. Prove that $\exists p \in F[x, y], p \neq 0 : p(f(t), g(t)) = 0$ I'd thank any hints that point me in the right direction.
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2answers
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Proofs involving endomorphisms on the space of polynomials

Define endomorphisms $D$ and $E$ on the space of polynomials with rational co-efficients $ \mathbb{Q}[x] $ such that $ D(x^n)= nx^{n-1}, E(x^n) = \frac{1}{n+1}x^{n+1} $ We must show that $ DE = I $ ...