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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Simplifying polynomials

Suppose I have a (multivariate) polynomial with coefficients in $\mathbb Z$ or $\mathbb Q$, given in fully expanded form. How can I simplify this to reduce the number of elementary operations ...
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Properties of Roots of polynomials

Today in highschool we were doing a chapter called "Roots of polynomials" where we learnt something new and interesting which is : $ax^2+bx+c=0$ Has roots $\alpha$ , $\beta$ Then: $$\alpha + ...
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1answer
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Space of real polynomials of one variable isn't complete

Consider $E$ - the vector space of all real polynomials of one variable. I need to prove that it is not complete under any norm. I was thinking I could use the fact that certain functions, for ...
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infinite limit question from Calc I

Find the limit $$\lim_{x\to\infty}\sqrt{x^2+x+1}-x$$ This limit is part of a question involving squeeze theorum, the limit is $\frac12$ but i don't know how to prove it because of the polynomial in ...
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2answers
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What is the general formula for power series summation? [duplicate]

While reviewing definite integrals, $\int_a^bf(x)dx$; I recalled that a definite integral could not only be solved by the difference of the anti-derivatives of intervals b and a, $F(b)-F(a)$, via the ...
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1answer
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Divisibility of polynomials in a subfield of a field.

I am trying to prove the following assertion: Let $K\subset L$ be fields, let $f,g\in K[x]$ be such that $f\mid g $ in $L[x]$, then $f\mid g$ in $K[x]$. We clearly have that $fh=g$ for some ...
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4answers
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how tp find a polynomial when the leading coefficient and some remainder are given?

The leading coefficient of a polynomial $P(x)$ of degree $3$ is $2006$. Suppose that $P(1)=5$, $P(2)=7$ and $P(3)=9$, then find $P(x)$. $(1)\ \ 2006(x-1)(x-2)(x-3)+2x+3$ $(2)\ \ ...
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1answer
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How to find the value of $m$ which is a power of a polynomial, when divided by a linear polinomial gives some remainder?

Q) The value of $m$ if $2x^m+x^3-3x^2-26$ leaves remainder of 226 when divided by $x-2$. (1) 0 (2) 7 (3) 10 (4) all of these How i solved it let $p(x)=2x^m+x^3-3x^2-26$ and ...
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2answers
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If $p,q$ are prime, $q$ odd $p \not\equiv 1 \pmod q$, is there an integer $x$ such that $p\mid 1+x+\ldots+x^{q-1}$

Suppose $p,q$ are two distinct prime numbers, $q \geq 3$ and $p \not\equiv 1 \pmod q$. Then I have the following problem: Prove that there is no integer $x \in \mathbb{Z}$ such that ...
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Proving $\int_0^1 B_n(x) dx=0$ for Bernoulli polynomials

The Bernoulli polynomials $B_k(.)$ are given by $$ \frac{t\:e^{xt}}{e^t-1}=\sum_{k=0}^\infty B_n(x)\frac{t^n}{n!}, \quad |t|<2\pi. \tag{1} $$ I would like to prove that $$ \int_0^1 B_n(x) dx=0, ...
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3answers
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Polynomials over $\mathbb{F}_2$ with certain values in $\mathbb{F}_4$

Let $\mathbb{F}_4=\{0,1,u,u^2\}$ be the field with $4$ elements. Is there a polynomial $p \in \mathbb{F}_2[x,y]$ with the following property? (1) For $r,s \in \mathbb{F}_4$, we have $p(r,s)=u ...
2
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2answers
27 views

Equal sums of powers

I want to show that if I have $2n$ numbers $a_1,a_2,\dots ,a_n,b_1,b_2,\dots b_n$ such that $\displaystyle \sum_{i=1}^{n}(a_i)^j=\sum_{i=1}^{n}(b_i)^j$ for every $j=1,2,\dots ,n$, then ...
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1answer
26 views

Third degree polynomial with integer coefficient from which one is odd has no integer roots

Third degree polynomial with integer, positive coefficient is given. Second coefficient is odd, other are even. how to prove that this polynomial has no integer roots
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1answer
48 views

If $k[X]/f = k[X]/g$, does $f = g$?

Let $k$ be a field and $f, g$ be irreducible monic polynomials in $k[X]$. Suppose $k[X]/f \stackrel{\sim}{=} k[X]/g$. Then does $f = g$? If so, how can this be generalized? Otherwise, how should I ...
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0answers
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Please clarify the following polynomial problem

Let $x_n$ be the remainder when $x$ is divided by $n.$ For positive integer $x$, compute the sum of all elements in the solution set of: $$x^5(x_5)^5 - x^6 - (x_5)^6 + x(x_5) = 0.$$ I just don't ...
2
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1answer
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Difficult Polynomial Question

Let $P(x)$ be a polynomial whose degree is 1996. If $P(n) = \frac{1}{n}$ for $n = 1, 2, 3, . . . , 1997$, compute the value of $P(1998).$ I don't even know where to begin... Any and all help would ...
3
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1answer
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Find the largest value of x given the equation…

Find the largest value of $x$ for which $x^2 + y^2 + z^2 = x + y + z$. What I did was subtract the RHS, to get $$x^2 - x + y^2 - y + z^2 - z = 0$$ $$x^2 - x + \frac{1}{4} + y^2 - y + \frac{1}{4} + ...
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1answer
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How do you determine where a polynomial evaluates to a perfect square?

How do you determine where a polynomial evaluates to a perfect square? One example would be $f(x)=x^2+148x-288$. $f(68) = 14400 = 120^2$. Another one would be $f(x)=x^2+204x-88$. $f(2) = 324 = ...
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0answers
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A polynomial in $\Bbb C[x, y, z]$ is irreducible

How to show that $x{^5}+y{^7}+z{^1}{^1}$ is irreducible in $\Bbb C[x, y, z]$? Give me some idea to solve it. Thank you.
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1answer
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Polynomials and endomorphisms

Let $E$ be a real vector space of finite dimension $n$ and $f \neq 0$ an endomorphism of $E$. I proved that there exist a real polynomial $P$ such that $P(f) = 0$. Now, we assume that $P$ has no real ...
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How to solve this equation in radicals?

How to solve the equation $x^6-2\varphi^5x^5+2\varphi x+\varphi^6=0$ in radicals? where $\varphi$ is the golden ratio.
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1answer
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Show that inequality is correct for natural $n$

Show that the following inequality is correct for all natural $n$ : $$(2n+1)^n\geq(2n)^n+(2n-1)^n$$ I've tried throwing the $(2n-1)^n$ or $(2n)^n$ on the left side and using formula of subtraction ...
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1answer
84 views

Some challenging Series, maximum value and polynomial factor questions

So I realize that the questions I am gonna ask are going to be a minute's work for some of you but I couldn't do them even after hours of searching for methods or something. They are from a ...
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1answer
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Factorization in noetherian domains

I changed the title (and the body) of this question page, since user26857 provided a nice answer for my original question in a more general setting. Here's what the accepted answer below provides: ...
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0answers
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Can you interpolate my polynomials if I give you some randomized values

Scenario (1) We define the polynomial ring $R[x]$ consist of all polynomial with coefficients from $\mathbb Z_p$, where $p$ is a prime number. Let $P_i$ be a polynomial such that $P_i \in R[x]$. We ...
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1answer
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Non-surjective but injective real polynomial functions $\mathbb{R}^n\to \mathbb{R}^n$

Over algebraically closed fields $K$, the Ax–Grothendieck theorem (see also this thread) states that injective polynomial functions $K^n \to K^n$ in $n$ variables are surjective. Is there a simple ...
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1answer
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Reasoning about Degree of Polynomial

How can you show that f is a polynomial of degree ≤ 2 if and only if its Hessian is constant
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A simpler method to show $x^6+x^3+1$ is irreducible in $\mathbb{Q}\left[x\right]$.

The original is show that $x^6+x^3+1$ cannot be written as a product of two polynomials of integer coefficients and positive degrees.(I think it is equivalent to show that the polynomial is ...
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1answer
42 views

How do you prove this theorem?

The theorem I have to prove is ...
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1answer
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Factor Theorem given two factors

The function $f(x)= ax^3-x^2+bx-24$ has three factors. Two of these factors are $x-2$ and $x+4$. Determine the values of a and b and then solve for $f(x)$. Please give an algebraic solution.
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Random multipliers of polynomial values at known points in $\mathbb{Z}_p$

Scenario (1) We define the polynomial ring $R[x]$ consist of all polynomial with coefficients from $\mathbb Z_p$, where $p$ is a prime number. Let $P_i$ be a polynomial such that $P_i \in R[x]$. We ...
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0answers
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Monotone increasing polynomial

Say I have a polynomial $$f(x)=\sum a_ix^i$$ where all the $a_i\geq 0$. Is there an easy way to see that it is monotone increasing?
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1answer
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Monotonically increasing polynomial intersection

Say that I have two strictly monotonically increasing polynomials of the form $f(x)=\sum^N_{i=0} a_ix^i, g(x)=\sum^N_{i=0} b_ix^i,$ with all coefficients $\geq 0$. Does it follow that they only ...
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1answer
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sum of square root of primes 2

I dont know how to solve the problem below. (1) $p[1]$, $p[2]$, $\ldots$, $p[n]$ are distinct primes, where $n = 1,2,\ldots$ Let $a[n]$ be the sum of square root of those primes, that is, $a[n] = ...
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2answers
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Implications of zero elemntary symmetric polynomials over a finite field

For a prime $q$ and an integer $n<q$, consider working over the finite field of $q^n$ elements. Denote by $s_n^k$ the $k$-th elementary symmetric polynomial in $n$ variables. That is, ...
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Terminology for polynomials

A multivariate polynomial in which the monomials have the same degree $d$ is called a homogeneous polynomial. An example for this is $$ 11x^2y + 2y^3 + 9yz^2 + xyz.$$ Is there a notation for ...
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How to expand such expression?

There is an array $a_n\in K$ and mapping $f:K\rightarrow K$ with properties: $f(a_0)=a_0$ $f(a_n)=a_{n+1}-na_{n-1}$ ask how to expand $f^{(m)}(a_0)=f(f(f(f...f(a_0)...)))$ $\quad$ (do $f$ operation ...
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0answers
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How to prove polynomial $p_{10}(z)$ has only real and negative zeros?

I encountered a polynomial: $$p_{10}=0.742134 + 32.583720 z + 345.639505 z^2 + 1369.404360 z^3 $$ $$+ 2400.069657 z^4 + 1996.926314 z^5 + 798.801952 z^6 + 147.695904 z^7 $$ $$+ ...
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1answer
37 views

prove that the given polynomial is irreducible polynomial over Q

If $a_0+a_1 x+ \ldots + a_ n x^ n$ is irreducible over $\mathbb Q$ then $a_n+ a_{1} x^{n-1}+\ldots+a_0$ is irreducible over $\mathbb Q$.
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1answer
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Suppose that p(x) is any polynomial in x with positive coefficients. Show that log(p(x))∈O(logx).

Suppose that p(x) is any polynomial in x with positive coefficients. Show that $log(p(x))∈O(log\ x)$. My professor posed this question in class today, and I'm not sure how to go about proving it. ...
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Iteration of a function related to the minimal polynomial of a matrix

Let $M$ be a singular $n \times n$ matrix over some field. In order to find a matrix $N$ s.t. $MN=0$, I do the following : $p(x)=$ minimal polynomial of $M$. Then the constant term of p is zero ...
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1answer
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A question about co-prime polynomials in $\Bbb{C}[x,y]$

Say $f$ and $g$ are two co-prime polynomials in $\Bbb{C}[x,y]$. Can the following expression always be written $$af+bg=1$$ where $a,b,f,g\in\Bbb{C}[x,y]$? I realise that the Euclidean algorithm is not ...
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1answer
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Proof of limit of polynomial function.

This exercise is used many places. Recall that a polynomial of degree $n$ is a function of the form $$P(x) = a_n*x^n + a_{n-1}*x^{n-1} + ....+ a_1x + a_0,$$ where $a_j$ are real numbers for $j = ...
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2answers
257 views

Interpolation polynomial Challenge

suppose $p(x)=x^k-x^t, k \neq t $ (k,t is a positive integer). function q(x) be a Interpolation polynomial from degree lower or equal n, to data $i=1,...,n+1, (x_i ,p(x_i))$. if ----------- then ...
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1answer
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Polynomials with coefficients $1$ or $2014$ [closed]

Let $$ P(x)=a_mx^m+a_{m-1}x^{m-1}+ \cdots+a_1x+a_0$$ and $$\quad Q(x) =b_nx^n+b_{n-1}x^{n-1}+ \cdots+b_1x+b_0 $$ be two polynomials where $a_i,b_j \in \{1,2014\}$ for all $i,j$. Suppose that ...
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1answer
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Pre-Calc question, please help.

I have to find a polynomial function of degree 4 with real coefficients with a real zero at i, a zero at -3(multiplicity 2), that passes through the point (-1,16). Please help.
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Roots of this third degree polynomial

I've got the following polynomial $$ x^3-6x^2-2x+40 $$ and I want to find its roots. The only option I see at the moment is to compute all the divisors of $40$ and their inverse, and manually check if ...
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3answers
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Chebyshev's Theorem regarding real polynomials: Why do only the Chebyshev polynomials achieve equality in this inequality?

In the book Proofs from The Book by Aigner and Ziegler there is a proof of 'Chebyshev's Theorem' which states that if $p(x)$ is a real polynomial of degree n with leading coefficient $1$ then $$ ...
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1answer
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How to find the polynomial which has the sum of two cube roots as one of its roots?

For example. How do I find the polynomial which has $\sqrt[3]2 + \sqrt[3]3$ as one of its roots? ( Hint: polynomial is $x^9-15x^6-87x^3-125$ )
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1answer
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Word problem for polynomial equations: Volume of a cylinder given relation between radius and height

A cylinder has a volume of 324 cm^3. If the radius of the cylinder is 1 cm more than twice the height, find the dimensions of the cylinder. I know that formula for the volume of cylinder is $V=\pi ...