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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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 ...
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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 ...
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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 ...
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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) &= ...
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260 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 ...
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2answers
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How do I take the 100th derivative of a polynomial [on hold]

How could I find $$f^{100}(x)$$ for $$f(x)=2x^{100}-7x^{80}+15x^{60}-27x^{40}-18x^{20}+300$$
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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: ...
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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 ...
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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 ...
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1answer
32 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 ...
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1answer
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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 ...
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3answers
63 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 ...
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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, ...
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1answer
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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 ...
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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 ...
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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 ...
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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 ...
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3answers
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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 ...
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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 - ...
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2answers
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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 $3xy^2$?
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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 ...
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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 ...
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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 $\epsilon$? ...
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2answers
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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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74 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 ...
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1answer
45 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 ...
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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 ...
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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{ ...
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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 ...
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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 ...
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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 ...
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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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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 ...
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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 ...
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1answer
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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 ...
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1answer
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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 ...
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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 ...
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1answer
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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. .
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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 ...
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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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1answer
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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 ...
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37 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 ...
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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 = ...
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Linear Algebra - Weighted Inner Product of Polynomials [closed]

Given the weighted inner product $\langle f,g\rangle = \int^1_{-1}f(x)g(x)x^2dx$ How do you find an orthogonal basis of the space $\Bbb P^1$ of polynomials of degree $\le$ 1. And how do you find the ...
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1answer
21 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?
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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 ...
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1answer
18 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 ...
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42 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 $y=5x+x^2$, ...
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44 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 ? ...
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Linear Algebra: Polynomials Basis

Consider the polynomials $$p_1(x) = 1 - x^2,\;p_2(x) = x(1-x),\;p_3(x) = x(1+x)$$ Show that $\{p_1(x),\,p_2(x),\,p_3(x)\}$ is a basis for $\Bbb P^2$. My question is how do you even go about proving ...