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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55 views

Proving $V(f)$ is finite when $f$ is non-constant

My problem asks me to show that if $f$ is non-constant, then $\mathbf{V}(f)$ is finite. Assume that $f \in \mathbb{C}[x]$. If $f$ was an ideal, this would be straightforward; however, $f$ is merely ...
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0answers
60 views

How prove this $\int_{0}^{1}P(x)dx>C_{n}$

Question: let the Polynomials $$P(x)=x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\cdots+a_{1}x+a_{0}$$ show that: there exsit $C_{n}$( only dependent on $n$ )such $$\int_{0}^{1}P(x)dx>C_{n}$$ ...
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3answers
26 views

Prove GCD of polynomials is same when coefficients are in a different field

Prove that the greatest common divisor of two polynomials $f, g$ in $\Bbb Q[X]$ is equal to their greatest common divisor in $\Bbb C[X]$. I am having trouble writing this proof. I tried setting it up ...
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2answers
26 views

How to show if it is irreducible

Let R be the subring (with identity) of Q[x, y] generated by x^2 , y^2 , and xy. Each of these elements is irreducible in R How do we know if they are irreducible
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Why doesn't Tutte polynomial T(1,1) equal 0?

If the formula for a Tutte polynomial is: then how does T(1,1) solve for spanning trees instead of just returning a 0?
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32 views

Finding polynomial's to satisfy a polynomial function

If I have that $p_{k,j}: F \to F$ is given by: $p_{1,0}=(x-2)^3$ $p_{2,0}=(x-1)$ and I have a polynomial function $f_0(x): F \to F$ given by $f_0(x)=1$ How would I find polynomials $h_1$ and $h_2$ ...
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1answer
60 views

Quotient of polynomial ring in two variables is a PID

With $K$ a field and $K[x,y]$ the polynomial ring over it in two variables, the quotient ring of it over the ideal generated by $1-xy$ is a PID. I've tried using Noetherianess but haven't gotten ...
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+50

On the problem of polynomial bijection from $\mathbb Q\times\mathbb Q$ to $\mathbb Q$

The question titled "Polynomial bijection from $\mathbb Q\times\mathbb Q$ to $\mathbb Q$" which was posed on MathOverflow attracted quite a lot of attention (and may be the question with most wrong ...
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1answer
40 views

How do I solve for the zeros of a Chebyshev polynomical? (on a computer)

I am working on a computer program and have a method that returns a number for a given $x$, $y$. So $f(x, y) = z$, where $f$ is my method. if I know $y$ and $z$, can I find what $x$ will be, without ...
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22 views

Factor $x^2+2x+2$ in $\mathbb{F}_3/(x^2+1)$

I am asked to find two roots of $x^2+2x+2$ in $\mathbb{F}_3[x]/(x^2+1)$ (the Kronecker construction). The elements of that field are (equivalence classes of) constant or linear polynomials in ...
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1answer
63 views

Real roots of $p(x)=x^n+ax+b$

What can we say about the real roots of $p(x)$? My Work: If $n$ is odd I found that $p$ has at most $3$ real roots if $a<0$ and $p$ has at most $1$ real root if $a\geq 0$. How can I classify the ...
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2answers
49 views

how many distinct real zeros a function has

$f(x)= x^4+2x^3-2x^2+1$ How many distinct real zeroes does $f$ have? Is it two because it crosses the $x$-axis twice or am I completely wrong?
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1answer
40 views

If $f(z)$ is a polynomial function of degree $n \ge 2$, prove that the sum of the residues of $\frac{1}{f(z)}$ is zero

Let $f(z)=a_nz^n +a_{n-1} z^{n-1} +...+a_1z+a_0$ be a polynomial of degree $n \ge 2$. Prove that the sum of the residues of $\frac{1}{f(z)}$ is zero. Ok, so here is my thinking process so far: At ...
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1answer
21 views

Finding an orthogonal basis of W and Proving that it is a subspace of P

$P(2)$ is the space of polynomials of degree $\le2$ with the inner product defined by $$(a_0 + a_1x + a_2x^2 , b_0 + b_1x + b_2x^2) = a_0b_0 + a_1b_1+ a_2b_2$$ and $P(x) = x^2 + x + 1$ and $W = ...
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1answer
33 views

A problem related to complex polynomial

Let P be a polynomial of degree n. Assume that |P(z)| ≤ M for |z| = 1. Show that |P(z)| ≤ M|z|^n for |z| ≥ 1. I don't know how to begin on this problem.
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2answers
36 views

An isomorphism between $\mathbb R^2$ and the space of polynomials of degree at most $1$

Let $V$ be the vector space of real-valued polynomials $p(x)$ of degree at most $1$. Prove that the map $$T:V\to\mathbb R^2,\quad p \mapsto [p(1), p(2)]$$ is an isomorphism. Progress For ...
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0answers
51 views

Roots of a polynomial equation where coefficients follow a geometric progression

Given a positive constant $a\in\mathbb{R}$, , and a positive integer $n$, I am interested in the roots of $x^n + \sum_{i=0}^{n-1} a^i x^{n-i-1} = x^n + x^{n-1} + a x^{n-2} + a^2 x^{n-3} +\cdots + ...
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1answer
34 views

Finding nth derivative of an exponential function and its value at the origin.

I have a function defined as $f(x) = e^{-\frac{1}{x^2}}, $if $ x\ne0$; $0$ if $x =0$. where $f:[0,\infty) \to \mathbb{R}$ I am asked to prove the following: (a) that the nth derivative is of the ...
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1answer
12 views

Another way to prove that if n$^{th}$-degree polynomial $r(z)$ is zero at $n+1$ points in the plane, $r(z)\equiv 0$?

The original problem is as follows Let $p$ and $q$ be polynomials of degree $n$. If $p(z)=q(z)$ at $n+1$ distinct points of the plane, the $p(z)=q(z)$ for all $z\in \mathbb{C}$. I attempted ...
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79 views

$f'(a)=0$ implies $x=a$ is not a simple zero of $f$

Let $a$ be the root of a polynomial $f(x)$ and let $f'(a)=0$. Then $x=a$ is not a simple zero of $f(x)$. What is the name of this theorem and does someone know a simple (high school level) proof?
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If $x^a + x^b = x^c + x^d$ how do $a ,b , c , d$ relationship are?

I used to solved these equation style and it's accidentally found an answer from matching $a, b, c,$ and $d$ relationship when $x^a + x^b = x^c + x^d $ (I assume that $ab = cd$) and found that's ...
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3answers
52 views

Solving $3t^2-\frac{12}{3}t+\frac{4}{3}=0$

I need to to solve: $$3t^2-\frac{12}{3}t+\frac{4}{3}=0$$ The solution manual factorizes this to $\dfrac{1}{3}(3t-2)^2$. How can you do this easily?
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0answers
62 views

How prove this polynomials $p_{j}(x)$ and $p_{k}(x)$ are relatively prime (2014,Putnam problem)

Question: Let $P_n(x)=1+2x+3x^2+\cdots+nx^{n-1}.$ Prove that the polynomials $P_j(x)$ and $P_k(x)$ are relatively prime for all positive integers $j$ and $k$ with $j\ne k.$ this problem it seem ...
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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
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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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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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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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2answers
130 views

Miklos Schweitzer 2014 Problem 8: polynomial inequality

Look at problem 8 : Let $n\geq 1$ be a fixed integer. Calculate the distance: $$\inf_{p,f}\max_{x\in[0,1]}|f(x)-p(x)|$$ where $p$ runs over polynomials with degree less than $n$ with real ...
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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
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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1answer
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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12answers
156 views

Simplifying $\frac{x^6-1}{x-1}$

I have this: $$\frac{x^6-1}{x-1}$$ I know it can be simplified to $1 + x + x^2 + x^3 + x^4 + x^5$ Edit : I was wondering how to do this if I didn't know that it was the same as that.
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1answer
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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
159 views

Show all roots of $\sum_{k=0}^n 2^{k(n-k)} x^k$ are real (December 6, 2014 Putnam problem)

Show that for each positive integer n, all roots of the polynomial $\sum_{k=0}^n 2^{k(n-k)} x^k$ are real numbers. I have no idea where to start. From this year's Putnam, problem B4.
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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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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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2answers
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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 ...
3
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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
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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5answers
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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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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60 views

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
32 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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63 views

Summation verification

I have a particular polynomial $$ 1-10x+35x^2-50x^3 $$ Which can be written nicely as $$1-(1+2+3+4)x+(1\cdot2+1\cdot3+1\cdot4+2\cdot3+2\cdot4+3\cdot4)x^2$$ ...
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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 ...
2
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5answers
76 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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2answers
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 ...
2
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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 ...