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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2
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2answers
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How to create a computationally cheap function passing through given points?

I am trying to develop a function which goes through the follow points. The function will be calculated on a microprocessor which has 20 mHz. List of given points: ...
20
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0answers
156 views
+50

Is There An Injective Cubic Polynomial $\mathbb Z^2 \rightarrow \mathbb Z$?

Earlier, I was curious about whether a polynomial mapping $\mathbb Z^2\rightarrow\mathbb Z$ could be injective, and if so, what the minimum degree of such a polynomial could be. I've managed to ...
-2
votes
2answers
47 views

How do I take the 100th derivative of a polynomial [on hold]

How could I find $$f^{100}(x)$$ for $$f(x)=2x^{100}-7x^{80}+15x^{60}-27x^{40}-18x^{20}+300$$
2
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0answers
31 views

Canonical algebra isomorphism $k[D(f)]\cong k[S_0,\dots,S_n]_{(f)}$?

Here's a common set up. Suppose you have $f\in k[S_0,S_1,\dots,S_n]$ is a homogeneous polynomial with $\deg(f)=d$, over some closed field $k$. Let $D(f)$ be the principal open set of $f$ in projective ...
2
votes
3answers
72 views

Find all intergers such that $2n^2+1$ divides $n^3+9n-17$

Find all intergers such that $2n^2+1$ divides $n^3+9n-17$. Answer : $n=(2 \ and \ 5)$ I did it. As $2n^2+1$ divides $n^3+9n-17$, then $2n^2+1 \leq n^3+9n-17 \implies n \geq 2$ So $n =2$ is ...
5
votes
3answers
99 views

Rules for whether an $n$ degree polynomial is an $n$ degree power

Given an $n$ degree equation in 2 variables ($n$ is a natural number) $$a_0x^n+a_1x^{n-1}+a_2x^{n-2}+\cdots+a_{n-1}x+a_n=y^n$$ If all values of $a$ are given rational numbers, are there any known ...
9
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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 ...
0
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0answers
10 views

Multitangent to a polinomial function

I'm trying to build some exercises on tangents of functions for beginner students in mathematical analysis. In particular I would like to suggest the study of polynomial functions $ y = p (x) $ of ...
15
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0answers
305 views
+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 ...
1
vote
1answer
28 views

Find a probability of $n$ event happening from $m$ types

The question is: to find a sum $$ S=\sum\limits_{n_1+n_2+\ldots+n_m = n,\ n_i=0,1,\ldots,n} p_1^{n_1}p_2^{n_2}\cdots p_m^{n_m}, $$ where $p_i\in[0,1]$. UPDATE. This issue has no probabalistic ...
1
vote
1answer
28 views

Diophantine equation : two products of linear factors differ by a constant

Recently, I was asked the following question by a friend : find all $a,b,c,a',b',c',k \in {\mathbb Z}$ with $k\neq 0$ such that the identity $$ (X-a)(X-b)(X-c)+k=(X-a')(X-b')(X-c') $$ holds in ...
0
votes
1answer
28 views

Having trouble combining Weierstrass approximation theorem and the infinite sequence of holomorphic functions

The Weierstrass approximation theorem says that all continuous functions on $[0,1]$ can be approximated uniformly by polynomials. Trying to facilitate the digestion of the fatty Christmas food, I ...
0
votes
3answers
59 views

Sum of Coefficients in a Polynomial

Find the sum of the coefficients of the terms in the expansion of $(2x+3y-3z)^7$. I know how to do this for binomials, but I was not able to apply the same logic to a trinomial. I believe my other ...
1
vote
1answer
29 views

Sum of Coefficients and Number of Terms in Trinomials and Quadrinomials

I already know how to find the sum of coefficients in a binomial, but how do you do it for a trinomial/quadrinomial (after like terms are added)? Example Problem: $(wa+xb+yc+zd)^n$ (all variables are ...
1
vote
3answers
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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0answers
34 views

What bounds can we put on the largest root of a polynomial?

For a polynomial $p(x)=x^{n+1}+a_{n} x^{n} + \cdots + a_1$ with roots $|x_1| < \cdots < |x_n|$ can we find relatively simple function $M(a_1, \dots, a_n)$ such that $$|x_i| \leq M(a_1, \dots, ...
0
votes
3answers
79 views

Discriminant of the polynomial $f(x)=4x^3-ax-b$

Definition. The discriminant of the polynomial $f(x)=4(x-x_1)(x-x_2)(x-x_3)$ is the product $16\{(x_2-x_1)(x_3-x_2)(x_3-x_1)\}^2$. How to prove that the discriminant of $f(x)=4x^3-ax-b$ is ...
0
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2answers
980 views

Graphing: Given two points on a graph, find the logarithmic function that passes through both.

Is there such a method to do this? I would like to come up with a logarithmic function (a graph that looks like a square root graph) that passes through two given points. Haven't had any luck in ...
2
votes
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 ...
0
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1answer
30 views

Sparse & Dense Polynomials

I've been reading up on Bernstein's theorem for an algebraic geometry assignment and I've come across the terms "dense" and "sparse" in relation to the polynomials. However, I have been unable to find ...
0
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0answers
15 views

What is the “cost” of computation of two special CAS algorithms

Suppose I have an integer $n$ with e.g. a large number of say decimal digits. I would like to get some information about the runtime "cost" of standard CAS algorithm which factors $n$ into primes ...
8
votes
1answer
204 views

How to solve $y^2=3x^4+3x^2+1$ for integers.

If $x,y \in \mathbb Z$ , then find all the solutions of $$y^2=3x^4+3x^2+1$$ I was asked this question by my friend who said that he encountered this while solving another problem. I have ...
5
votes
1answer
79 views

Polynomials with specified ranges in intervals

Say I have two finite intervals $[a,b],[c,d]\subsetneq\Bbb R$ where $a<b<c-1<c<d$ and $b-a=d-c=s<1$. I want to find a polynomial $f \in \Bbb R[x]$ such that $$\forall x\in[a,b],\mbox{ ...
0
votes
1answer
22 views

Determining how many roots a cubic equation has.

I am working through some of the quizes on brilliant.org I came across this question. Suppose that the following cubic polynomial has one rational root and two non-real complex roots: $$ x^3 - ...
0
votes
2answers
25 views

Multiplying with Polynomials.

In $(3xy)^2$, do I distribute that power of two to each of the terms? $(3^2)\times(x^2)\times(y^2) = 9x^2y^2$? Or do I just treat it as $3xy^2$?
0
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0answers
13 views

Where can I find material on polynomial filters?

Most students and mathematicians probably know a fair amount on roots-of-unity filters, or on Fourier analysis. The basic notion of this "filtering" is, given a polynomial, we can find the $n$th ...
3
votes
1answer
50 views

Writing solution to an arbitrary ODE with arbitrary initial values as the sum of a power series?

How can we solve for $y$ with these arbitrary initial values and polynomials? How would we write the solution as a power series?
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3answers
29 views

Lower bound for degree of polynomial.

Let $f:\mathbb{R}\to\mathbb{R}$ be a polynomial such that $$|f(x)|<\epsilon\quad\text{for all $x$ with }|x|<1.$$ Can we find an explicit lower bound for the degree of $f$ in terms of $\epsilon$? ...
9
votes
1answer
112 views

Product of numbers $\pm\sqrt{1}\pm\sqrt{2}\pm\cdots\pm\sqrt{n}$ is integer

Prove that the product of the $2^n$ numbers $\pm\sqrt{1}\pm\sqrt{2}\pm\cdots\pm\sqrt{n}$ is an integer. I want to consider the polynomial $P(x)=(x-a_1)(x-a_2)\cdots(x-a_{2^n})$, where the $a_i$'s are ...
2
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2answers
31 views

What is the difference between Algebraic Expressions and Polynomials?

Both are a combination of terms grouped together. What is the difference?
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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 ...
2
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0answers
66 views

homework: rings, matrices and polynomials

$A,B$ are both $n \times n$ and diagonal matrices. Prove that there is a matrix $X$ which is $n \times n$, and polynomials $p$ and $q$ such that $A= p(X), B= q(X)$ Is this true for ANY 2 matrices (we ...
2
votes
1answer
44 views

Is the polynomial a zero polynomial?

Let $p(x)$ be a polynomial over $\mathbb{R}$ with $deg[p(x)]\leqslant n$. If $p(1)=p(2)=\cdots = p(n+1)=0$, then will the polynomial be necessarily a zero polynomial? i.e., if a polynomial of degree ...
0
votes
2answers
73 views

How to factor polynomials in $\mathbb{Z}_n$?

How to factor a certain polynomial over $Zn$. for example factor the following polynomial into irreducible polynomials in $Z5$: $X^3+X^2+X-1$ or factor the following polynomial into irreducible ...
14
votes
3answers
193 views

Sum of $k$-th powers

Given: $$ P_k(n)=\sum_{i=1}^n i^k $$ and $P_k(0)=0$, $P_k(x)-P_k(x-1) = x^k$ show that: $$ P_{k+1}(x)=(k+1) \int^x_0P_k(t) \, dt + C_{k+1} \cdot x $$ For $C_{k+1}$ constant. I believe a proof by ...
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0answers
29 views

finding root of 3rd degree math equation

I need to solve the following equation and give a simple formula for $y$ such that with the known value of $x$ we can easily compute value of $y$. $$x = \frac{(c+ky)y^{2}}{2}$$ $c$ and $k$ are ...
0
votes
2answers
43 views

Can someone help me to solve the value of a,b,c,d? [on hold]

We have $$0.476=a(500)^3+b(500)^2+c(500)+d \\ 1.038=a(1100)^3+b(1100)^2+c(1100)+d \\ 1.982=a(2100)^3+b(2100)^2+c(2100)+d \\ 2.557=a(2700)^3+b(2700)^2+c(2700)+d \\ 3.240=a(3400)^3+b(3400)^2+c(3400)+d ...
53
votes
3answers
745 views

A real number $x$ such that $x^n$ and $(x+1)^n$ are rational is itself rational

Let $x$ be a real number and let $n$ be a positive integer. It is known that both $x^n$ and $(x+1)^n$ are rational. Prove that $x$ is rational. What I have tried: Denote $x^n=r$ and $(x+1)^n=s$ ...
3
votes
1answer
181 views

Testing polynomial equivalence

Suppose I have two polynomials, P(x) and Q(x), of the same degree and with the same leading coefficient. How can I test if the two are equivalent in the sense that there exists some $k$ with ...
0
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0answers
9 views

minimal polynomial of a LFSR sequence

I encircled the problem in the figure below. My question is, why $m(x)$ must have degree of at least $u$, why not it has a degree less than $u$? Maybe this is trivial, but I cannot wrap my mind ...
0
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0answers
13 views

positive Maclaurin polynomials

Consider even degree Maclaurin polynomials $M[n;2k]$ for $(1+x)^n$ where degree $= 2k < n$ and $n$ is a positive integer. Examples: (1) The quadratic #$M[3;2] = 1 + 3x + 3x^2$ is clearly ...
7
votes
4answers
703 views

Show $x^6 + 1.5x^5 + 3x - 4.5$ is irreducible in $\mathbb Q[x]$.

Show $p(x) = x^6 + 1.5x^5 + 3x - 4.5$ is irreducible in $\mathbb Q[x]$. By Gauss' Lemma, a primitive polynomial in $\mathbb Z[x]$ is irreducible in $\mathbb Q[x]$ if and only if it is irreducible in ...
9
votes
2answers
170 views

Artin Chapter 11, Exercise 9.12, polynomials without common zeroes [closed]

How do I show that the three polynomials $f_1 = t^2 + x^2 - 2$, $f_2 = tx - 1$, $f_3 = t^3 + 5tx^3 + 1$ generate the unit ideal in $\mathbb{C}[t, x]$? Artin mentions two approaches: by showing that ...
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1answer
30 views

For $f, g \in K[t]$, $f \neq g$ implies $f_K \neq g_K$

Consider an infinite field $K$. For $f, g \in K[t]$, show that $f \neq g$ implies $f_K \neq g_K$, where $f_K, g_K: K \rightarrow K$ denote the usual polynomial functions. My attempt: By ...
9
votes
1answer
114 views

Prove that $ ax^2+bx+c=0 $ has at least one root in $(0,1)$ if $10a+12b+15c=0$

If $10a+12b+15c=0$, Prove that $$ ax^2+bx+c=0 $$ has at least one root in $(0,1)$. Progress I tried to solve this by Rolle`s theorem ($f'$ has a root between any two roots of $f$), but could not ...
0
votes
1answer
19 views

Polynomial with even degree

suppose that P(x) is a polynomial with even degree and positive leading coefficient and that P(X) is greater than its second derivative.prove that P is non-negative
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1answer
23 views

Derivation: How do I derivate this

How do I deveriate the following expression? The problem I have is the n in d^n. This expression is part of a bigger task of mine : Show via complete induktion that is true for all n from ...
4
votes
2answers
111 views

Maximum absolute value of polynomial coefficients

Suppose we have a polynomial in integer coefficients $$p = p_0 + p_1 x + p_2 x^2 + \ldots + p_n x^n, p_k \in \mathbb{Z}$$ Now define $M(p)$ as the maximum absolute value of the coefficients of $p$, ...
2
votes
1answer
34 views

How do you find a basis for a polynomial in P2 given a set of polynomials?

I don't know how to show that p1, p2, and p3 actually form a basis for P2. I have been trying different things, but that fixed scalar c has prevented me from forming a basis. .
13
votes
1answer
133 views

Natural density of solvable quintics

A recent question asked about the topological density of solvable monic quintics with rational coefficients in the space of all monic quintics with rational coefficients. Robert Israel gave a nice ...