# Questions tagged [polynomials]

For challenges involving polynomials, mathematical expressions that consist of variables and coefficients.

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### Pretty Print Polynomials

Introduction Humans are a remarkable species, but we can be very awkward to understand sometimes—especially for computers. In particular, we seem to like writing polynomials in a very ...
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### Absolute Sums of Sidi Polynomial Coefficients

Background The Sidi polynomial of degree $n$ – or the $(n + 1)$th Sidi polynomial – is defined as follows. $$S_n(x) = \sum^n_{k=0}s_{n;k}x^n \text{ where } s_{n;k} = (-1)^k\binom n k (k+1)^n$$ The ...
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### Fundamental Solution of the Pell Equation

Given some positive integer $n$ that is not a square, find the fundamental solution $(x,y)$ of the associated Pell equation $$x^2 - n\cdot y^2 = 1$$ Details The fundamental $(x,y)$ is a pair of ...
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What is the Ultraradical? The ultraradical, or the Bring radical, of a real number $a$ is defined as the only real root of the quintic equation $x^5+x+a=0$. Here we use $\text{UR}(\cdot)$ to ...
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### Многочлены Чебышёва (Chebyshev Polynomials)

Chebyshev Polynomials are a family of orthogonal polynomials that pop up in all kinds of places in math, and they have a lot of quite interesting properties. One characterization of them is that they ...
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### Find real roots of a polynomial

Write a self-contained program which when given a polynomial and a bound will find all real roots of that polynomial to an absolute error not exceeding the bound. Constraints I know that Mathematica ...
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### Calculate Power Series Coefficients

Given a polynomial $p(x)$ with integral coefficients and a constant term of $p(0) = \pm 1$, and a non-negative integer $N$, return the $N$-th coefficient of the power series (sometimes called &...
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### Find the binarray!

We define a binarray as an array satisfying the following properties: it's non-empty the first value is a 1 the last value is a ...
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### Evaluate polynomial expression string

Create a function which takes a polynomial equation, a value for x and returns the result of the operation. Example: given ...
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### Are you a probabilist or a physicist?

Hermite polynomials refer to two sequences of polynomials: The "probabilist's Hermite polynomials", given by $${He}_n(x) = (-1)^n e ^ \frac {x^2} 2 \frac {d^n} {dx^n} e ^ {-\frac {x^2} 2}$$ ...
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### Polynomial Long Division

Implement polynomial long division, an algorithm that divides two polynomials and gets the quotient and remainder: (12x^3 - 5x^2 + 3x - 1) / (x^2 - 5) = 12x - 5 R 63x - 26 In your programs, you will ...
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### Generating generating expressions for sequences

(yes, "generating generating" in the title is correct :) ) Context In middle (?) school we are taught about sequences and, in particular, we are taught about linear sequences where the ...
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### Euler-Poincaré-Characteristic of Polyhedra

Given a triangulation of the surface of a polyhedron p, calculate its Euler-Poincaré-Characteristic χ(p) = V-E+F, where ...
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### Multiply Two Integer Polynomials

Your task is to take two single-variable integer polynomial expressions and multiply them into their unsimplified first-term-major left-to-right expansion (A.K.A. FOIL in the case of binomials). Do ...
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### Solve quadratic equations when 1+1=0

There already have been multiple challenges about carryless multiplication, this challenge will work with the same calculation rules. You task is given a quadratic polynomial ...
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### Print all Polynomials

The set of all polynomials with integer coefficients is countable. This means that there is a sequence that contains each polynomial with integer coefficients exactly once. Your goal is it to write a ...
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### Characteristic polynomial

The characteristic polynomial of a square matrix $A$ is defined as the polynomial $p_A(x) = \det(Ix-A)$ where $I$ is the identity matrix and $\det$ the determinant. Note that this definition ...
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### Resultant of two polynomials

The resultant of two polynomials is a polynomial in their coefficients that is zero if and only if $p$ and $q$ have a common root. It is a useful tool for eliminating variables from systems of ...
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### Algebraic curve plotter

An algebraic curve is a certain "1D subset" of the "2D-plane" that can be described as set of zeros {(x,y) in R^2 : f(x,y)=0 }of a polynomial <...
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### Exponential transform of an integer sequence

The exponential generating function (e.g.f.) of a sequence $a_n$ is defined as the formal power series $f(x) = \sum_{n=0}^{\infty} \frac{a_n}{n!} x^n$. When $a_0 = 0$, we can apply the ...