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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Quadratic polynomial in ${\Bbb C}$ that vanishes on three different points of a complex line

Here is my question: Let $f(x,y)\in {\Bbb C}[x,y]$ be a quadratic polynomial. If $f$ vanishes on three different points (say, $p,q,r$) of a complex line $$ L:=\{(x,y)\in{\Bbb C}^2\mid ...
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2answers
69 views

Linear functionals and dual bases

How do I tackle this question? I am a little hazy on linear functionals and integral signs.
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0answers
18 views

Matlab Code about evaluating Newton Polynomials

I am trying to write a code for evaluating a newton polynomial with coefficients $a = [a_0 , ... , a_n]$, and nodes $x = [x_0 , ... , x_n]$ at the vector $t$, using nested multiplication. ...
2
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0answers
39 views

Polynomial decomposition

I've just recently learned about the neat algorithm that, given a polynomial $f$ finds (non linear) polynomials $h,g$ such that $$f = g \circ h \quad (1),$$ or decides that there are no such ...
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1answer
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Herstein - Topics in Algebra - Polynomial rings page 157

In Chapter 3.9 of his book "Topics in Algebra" , 2nd ed, Herstein describes an example of a Quotient ring, namely $ F[x]/(x^3-2) = F[x]/A $ where $F = Q $ the rationals, and $(x^3-2) = A $ is the ...
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1answer
45 views

Karatsuba Method

For polynomials $f(x)$, $g(x)$ of degree $d = 2^{r-1}-1$, how do I check that multiplying $f(x)$ and $g(x)$ by the Karatsuba method requires $3^{r-1}$ multiplications in the field $F$?
3
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1answer
49 views

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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3answers
63 views

Factorize $6x^2 -5x -14 = 0$

I'm throwing a bit of a blank on the best way to factor this : $$6x^2 -5x -14 = 0$$ I know that I could multiply $6$ by $14$ and then find a pair of factors that add to $-5$ (b), but this feels a ...
0
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1answer
28 views

Simplifying the difference quotient $\frac{(x + h)^3 - x^3}{h}$.

For the function $f(x) = x^3$, I have the difference quotient: $$ \frac{(x + h)^3 - x^3}{h} $$ I tried changing the $(x + h)^3$ to $(x + h)(x^2 - xh + h^2)$ that I know to see if I could get ...
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3answers
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Show some polynomial is irreducible over the field of 7 elements.

I have to show that the polynomial $x^4+x^3+x^2+x+1$ is irreducible over the field $F_7$. It doesn't have roots in $F_7$, but I can't show it does not have degree two irreducible factors in $F_7[x]$. ...
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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
33 views

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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1answer
51 views

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
25 views

Algorithm complexity for roots of a polynomial [on hold]

For a polynomial in one variable of degree $n$, what is the algorithm complexity for root finding of the best/typically used algorithm?
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2answers
40 views

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
42 views

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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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 ...
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4answers
38 views

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
12 views

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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2answers
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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, ...
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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2answers
42 views

Help with completing the square [closed]

You have 240 feet of wooden fencing to form two adjacent rectangular corrals. You want each corral to have an area of 1000 square feet. Draw a picture that represent the corral, write an equation for ...
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2answers
103 views

How to Show Polynomial Growth < Exponential Growth (Without L'Hopital!)

Can anyone offer me a way to show that exponential growth trumps polynomial growth, without using L'Hopital's Rule? When I learned function growth speeds in high school, the closest thing to a proof I ...
0
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1answer
34 views

How to use Newton's method to find the roots of an oscillating polynomial? [closed]

Use Newton’s method to find the roots of $32x^6 − 48x^4 + 18x^2 − 1 = 0$ accurate to within $10^{-5}$. Newton's method requires the derivative of this function, which is easy to find. Problem is, ...
0
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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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2answers
240 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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5answers
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Check the proof of:$x^2+1$ is irreducible over $\Bbb Q$

Prove: $x^2+1$ is irreducible over $\mathbb{Q}$ Proof: Since $x^2+1=(x-i)(x+i)$, so $x^2+1$ is irreducible over $\mathbb{Q}$. Is it right?
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1answer
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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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1answer
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Please write out and rearrange this term: [closed]

1) Please fully write out this term: $$(W_p-W_{p-s})(W_q-W_{q-s})(W_r-W_{r-s})=\cdots$$ 2) Then please rearrange the term to: $$W_pW_qW_r-W_{p-s}W_{q-s}W_{r-s}+(\cdots)$$ Thank you.
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2answers
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Is there a rule to the terms of a falling factorial?

$\require{cancel}$I discovered that $n!=\xcancel{(n)_{n-1}}n^{\underline{n-1}}=n(n-1)(n-2)\cdots(3)(2)$. I have expanded a few examples: $$2!=\xcancel{(2)_1}2^{\underline{1}}=2\\ ...
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1answer
82 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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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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0answers
19 views

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
34 views

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
43 views

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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1answer
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There is a unique polynomial interpolating $f$ and its derivatives

I have questions on a similar topic here, here, and here, but this is a different question. It is well-known that a Hermite interpolation polynomial (where we sample the function and its derivatives ...
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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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0answers
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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
41 views

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
33 views

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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50 views

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
50 views

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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On the maximum number of polynomials in a certain subspace

I've already asked this question on mathoverflow, but no one answered. So I put this problem also here. Sorry for that. Let $\mathbb F_q$ be a finite field and let $e, k$ be positive integers with ...
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1answer
41 views

How do you prove this theorem?

The theorem I have to prove is ...
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1answer
35 views

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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2answers
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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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0answers
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Uniqueness of maximum derivative for rational function

Say I have a function $$f(x)=\frac{a_nx^n}{\sum_{i=0}^n a_i x^i}$$ where all the $a_i\geq 0$, $a_0=1$, and $a_n\neq 0$. Is there an easy but rigorous way to see that its maximum derivative is unique, ...
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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 $\mathbb Q$.