This tag is used for both basic and advanced questions on polynomials in any number of variables. Including, but not limited to: solving for roots, factoring, checking for irreducibility. This tag is rarely used as the only tag for a question.

learn more… | top users | synonyms

3
votes
4answers
93 views

AlgebraII factoring polynomials

equation: $2x^2 - 11x - 6$ Using the quadratic formula, I have found the zeros: $x_1 = 6, x_2 = -\frac{1}{2}$ Plug the zeros in: $2x^2 + \frac{1}{2}x - 6x - 6$ This is where I get lost. I factor $-...
0
votes
0answers
36 views

Is there a injective polynomial function from $R^2$ to $R$? [duplicate]

There is an injective polynomial function from $N^2$ to $N$ (the Cantor-pairing function for example, which is of degree 2), and also one of degree 4 from $Z^2$ to $Z$. I believe the question is open (...
3
votes
3answers
149 views

How can I prove that $g(\zeta)\in\mathbb R\implies g(\zeta)=h(\zeta+\bar\zeta)$

How can I prove that if $g(X)\in \mathbb Q[X]$ and $\zeta\in\mathbb C\backslash \mathbb R$, therefore $$g(\zeta)\in\mathbb R\implies g(\zeta)=h(\zeta+\bar\zeta)$$ for a certain polynomial $h(X)\in\...
1
vote
1answer
56 views

Basis for the space of quadratic polynomials orthogonal to those with $p(2)=p(1)$

Let $P_2[x]$ be the space of polynomials of degree less than or equal to 2. If $W = \{p ∈ P_2[x] \mid p(2) = p(1)\}$, then find a basis for $ W^⊥$ where $P_2[x]$ is equipped with an inner product ...
7
votes
1answer
51 views

Is it possible that $|z+\sum_{i\not=1} a_i z^i| <1$ for some $a_i \in \mathbb{C}$ and for all $|z|=1$?

I wonder that whether there exists a complex polynomial of the form $$ P(z)= z+\sum_{i\not=1} a_iz^i, a_i,z\in \mathbb{C},$$ (i.e. its first order term has coefficient 1) s.t. its modulus is less than ...
1
vote
2answers
54 views

Existence of complex polynomial with modulus on $|z|=1$ less than 1

I wonder if there exists a complex polynomial $P(z),z\in \mathbb{C}$ s.t $$\forall |z|\leq 1, P(z)<1.$$ I know that using modulus maximum principle, we only need to find $$P(z)<1, \forall |z|=...
1
vote
2answers
56 views

Showing $(Tp)(x) = x^2p(x)$ is a linear map (transformation)

Define a linear map function $T: \mathcal{P}(\mathbb{R}) \to \mathcal{P}(\mathbb{R})$ where $\mathcal{P}(\mathbb{R})$ is the set of all polynomials with real-valued coefficients. Now let $T$ belong to ...
9
votes
1answer
114 views

Replacing numbers by roots of quadratic

We have $10$ numbers in the interval $(0,1)$, not necessarily distinct. At any moment, we can choose two of them, $a$ and $b$. If the quadratic $x^2-ax+b$ has two (possibly identical) real roots, we ...
1
vote
2answers
45 views

Simplification ideas

Looking for a neat simplification idea to be able to solve for $x$ analytically in the expression below: $$S=k\tan x-Bk^2\frac{1}{\cos^2x}$$ where $\{S,k,B\}\neq0$ and $\in \mathbb{R}^+.$ Of ...
3
votes
0answers
37 views

Coefficients of Lagrange resolvent

I'm trying to make sense of some things I read about Galois theory. Let $p$ be a monic polynomial of degree $n$ with known coefficients $a_i$ and unknown roots $x_i$: \begin{alignat*}{2} p(X) &= (...
2
votes
5answers
488 views

Sum of roots: Vieta's Formula

The roots of the equation $x^4-5x^2+2x-1=0$ are $\alpha, \beta, \gamma, \delta$. Let $S_n=\alpha^n +\beta^n+\gamma^n+\delta^n$ Show that $S_{n+4}-5S_{n+2}+2S_{n+1}-S_{n}=0$ I have no idea how to ...
2
votes
3answers
244 views

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: ...
5
votes
1answer
138 views

Multitangent to a polynomial 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 ...
1
vote
1answer
48 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 ${\...
4
votes
3answers
90 views

Solve the equation $2x^2+5y^2+6xy-2x-4y+1=0$ in real numbers

Solve the equation $2x^2+5y^2+6xy-2x-4y+1=0$ The problem does not say it but I think solutions should be from $\mathbb{R}$. I tried to express the left sum as a sum of squares but that does not work ...
1
vote
1answer
96 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 ...
1
vote
3answers
88 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
327 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
1answer
40 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 ...
0
votes
3answers
112 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 $a^3-27b^...
0
votes
0answers
48 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 ...
2
votes
3answers
82 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 ...
0
votes
1answer
199 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
33 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 $3xy^2$?
90
votes
1answer
2k views

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 ...
1
vote
3answers
37 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 $\epsilon$?
2
votes
4answers
6k views

What is the difference between Algebraic Expressions and Polynomials?

Both are a combination of terms grouped together. What is the difference?
0
votes
2answers
123 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 ...
3
votes
1answer
55 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 $...
1
vote
0answers
50 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 ...
5
votes
1answer
101 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{ }...
-3
votes
1answer
48 views

systems of equations with one inequality and exponents.

I have a systems of equations for a website that relies on the solution. I have just read up on the subject online but I still can't come to a conclusion. The equation is this: $$8000 \lt xy^5 \lt ...
0
votes
2answers
50 views

systems of equations with exponents?

I am building a website which will run on the equation specified below. I am in pre-algebra and do not have any idea how to go about this equation. my friends say it is a system of equation but I don'...
1
vote
1answer
53 views

Polynomial with even degree

Suppose that $P(x)$ is a polynomial with even degree and positive leading coefficient and $P(x)\ge P''(x)$. Prove that $P$ is non-negative.
12
votes
1answer
165 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 ...
9
votes
1answer
149 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 ...
1
vote
1answer
42 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 ...
1
vote
1answer
35 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 Euclidean ...
1
vote
1answer
163 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 ...
2
votes
1answer
483 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. .
1
vote
2answers
74 views

Find the value of P(2014) given some properties about this polynomial…

A polynomial P satisfies the following criterion: It's coefficients are integers. For all real $(a, b, c, d)$ we have $(P(a) + P(b))(P(c) + P(d)) = P(ac - bd) + P(ad + bc)$. Determine all possible ...
13
votes
3answers
215 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 ...
0
votes
1answer
50 views

How to take apart a characteristic polynomial

Suppose I have a polynomial: $x^3-8x^2+17x-4$. How do I know it will always be $(x-4)(x^2-4x+1)$ by solving it? I'm struggling to figure out what to look for in the polynomial to give me a hint or ...
0
votes
0answers
39 views

Calculate the product of $p(x)q(x) \pmod{x^3 +1}$

I need to calculate the product of $(x^2 + 3x + 1)(x^2 + 4x + 3)\pmod{x^3 + 1}$, where the product is in $\mathbb{Z}_5[x]$. Is this problem as simple as just multiplying the two, which would be $4x^4 ...
1
vote
2answers
102 views

Trying to understand a proof for the automorphisms of a polynomial ring

Consider the following proof for finding all automorphisms of the ring $\mathbb{Z}[x]$ which I am trying to understand. Source I have two question regarding the proof 1) They set $d = \deg(\phi(f(x)...
0
votes
1answer
26 views

Inverse of a polynomials

The polynomial $f(x)=2x+1\in\mathbb{Z}_{4}[X]$ have inverse in the ring $\mathbb{Z}_{4}[X]$? How to determine this polynomial?
1
vote
2answers
45 views

How to solve for $y$ on five equations

It has been over ten years since I've taken an algebra course so I'm sure I am doing something simple incorrectly. I have a series of five equations. Given a specific $x$ value (body weight) I want to ...
0
votes
1answer
52 views

Linear Algebra - Inner Products, Functions, and Closet Polynomial

This is the question: Formulate the linear algebra problem of finding the closet poly $p \in span \{1, t^2\}$ to the function $f(t)=e^tcos(t)$ with respect to the L$^2$ inner product: $\lt f,g\gt =...
4
votes
1answer
55 views

(Though?)Expression Rearranging

I have the following expression $ 2x+3x^2+e^{5x+x^2}=7 $ which I need rearranged in a form of the type $Ye^Y=Z$ with Y a function of x and Z some constant. I have tried the substitution $y=5x+x^2$, ...
1
vote
0answers
89 views

Real roots of an nth order polynomial

Given an nth order polynomial, is there any algorithm that can calculate all the roots ? Is there any algorithm that can calculate ALL the roots of the equation ? $$p(x)=p_nx^n+p_{n-1}x^{n-1}+\cdots+...