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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Solving the polynominal: $s(t) = -16t^2 + 48t + 160$

The height of a ball is thrown directly upward from an initial height of $160$ ft with an initial velocity of $48$ ft per second is given by the function: $s(t) = -16t^2 + 48t + 160$, where $s(t)$ ...
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1answer
37 views

How to find the solution of a quadratic equation with complex coefficients?

I know how to find the solution for a quadratic equation with real coefficients. But if the coefficient changes to complex numbers then what is the change in the solution? Want an example of such ...
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1answer
46 views

Kantorovich Theorem example

I need to write in C a program that finds roots of a 6th order polynomial. I was thinking of using Kantorovich Theorem convergence of Newton's method to find when can I use Newton-Rhapson method. I'm ...
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1answer
55 views

I need help proving a statement about rational roots

I have no idea where to start...this is the statement: If a polynomial of degree not greater than 5 with rational coefficients has multiple roots, it has also a rational root, except in the case ...
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34 views

Decomposition of polynomials and inequality

This was asked in comment here by @23rd : If $f$ is a polynomial with $\deg f=n\ge2$, then there exist polynomials $g$ and $h$, such that $$f(x)=2xg(x)−h(x)$$ $$\deg g\le n−1, \quad \deg h \le ...
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Polynomial long division to determine the quotient [closed]

Use polynomial long division to determine the quotient when 3x^3 - 5x^2 + 10x +4 is divided by 3x + 1?
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36 views

Construct a polynomial with a certain root.

Suppose we have a polynomial $g(x)=ax^3-bx^2+cx-d\in\mathbb{Z}[x]$ whose zero is $\rho$. How do we construct a polynomial which involves $g(x)$ and with a root $\frac{d}{\rho}$? I was trying to do ...
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5answers
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Polynomials Shouldn't Have factors using Rational Root Theorem but it does!

I came across this polynomial $X^4 + X^3 + 2X^2 + X + 1$ I tried to factor it using Rational root theorem, but it seems there are no roots possible. 1 or -1 don't work. But I know for a fact ...
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solving the inequalty

are there any ways to solve :$ x^4 -6x^3 +28x^2 -64x +96 >0$ ?
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17 views

Sufficient condition for a indefinite integral to be an elementary function

I would like to find a sufficient condition on two polynomials $P(s)$ and $Q(s)$, such that the function $s \mapsto Q(s)e^{P(s)} $ has a primitive integral of the form $s \mapsto R(s)e^{P(s)} $ (with ...
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1answer
26 views

remainder is not zero using long division method

Find all zeros of $f(x)=128x^3-48x^2+1$ given that one linear factor occurs twice. let $f(x) $ be equaal to 0 $128x^3-48x^2+1=0,$ $16x^2(8x-3)+1=0,$ trying $x=1/4$ $16/16(2-3)+1=0,$ ...
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2answers
77 views

Most Efficient Method to Find Roots of Polynomial [duplicate]

I am designing a software that has to find the roots of polynomials. I have to write this software from scratch as opposed to using an already existing library due to company instructions. I currently ...
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1answer
40 views

Determine the coefficients of a polynomial knowing its roots

My prof. gave this problem as a bonus in an exam, and I couldn't figure out a solution. Some hints or a general method of solving it would be very nice. Given the following polynomial: ...
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1answer
31 views

Polynomials vanishing on subsets of $\mathbb{R}^2$

Let $\mathcal{S}\subset\mathbb{R}^2$ such that every point in the real plane is at most at distance $1$ from a point in $\mathcal{S}$. Is it true that if $P\in\mathbb{R}[X,Y]$ is a polynomial that ...
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0answers
32 views

Set of Metapolynomials is closed under multiplication

We say that a function $f:\mathbb{R}^k \rightarrow \mathbb{R}$ is a metapolynomial if, for some positive integer $m$ and $n$, it can be represented in the form $$f(x_1,\cdots , x_k ...
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4answers
109 views

Finding double root of $x^5-x+\alpha$

Given the polynomial $$x^5-x+\alpha$$ Find a value of $\alpha>0$ for which the above polynomial has a double root. Here's an animated plot of the roots as you change $\alpha$ from $0$ to $1$ I'm ...
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0answers
37 views

Extension of quadratic forms

A homogen multivariate polynomial with degree 2 is a quadratic form. It can be checked if the polynomial is positive for any non-zero vector by checking if the corresponding matrix A is positive ...
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1answer
45 views

What is the Most Efficient Way to Calculate the Internal Rate of Return?

I have built a program that prices financial assets and it does this in part by calculating the IRR. The problem is that it does not run as quickly as I would like it to. I currently use the ...
3
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1answer
153 views

An integral and $\pi(n)$

Are there polynomials $P,Q\in \mathbb{R}[x]$ satisfying : $$\int_{0}^{\log n}\frac{P(x)}{Q(x)}\,\mathrm{d}x=\frac{n}{\pi(n)}\quad \text{ for infinitely many }n\in \mathbb{N}$$ Here $\pi(n)$ is the ...
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5answers
47 views

Fraction with negative exponent fraction.

Q: $$\left(\frac{27 a^6 b^{-3}}{c^{-2}}\right)^{-2/3}$$ A: $$\frac{b^2}{9 a^4 c^{4/3}}$$ How in the world are they getting that?
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evaluation of polynomial regression

I have a data set $(x_i$ $y_i)$ if=1...20. I have to fit the data using polynomial feature. How can I evaluate what the complexity of model should be chosen? There is a hint in the task using RMSE ...
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1answer
23 views

Explicit delta for polynomial limit

I'm looking for an explicit formulation for $\delta$ in the $\epsilon-\delta$ formulation of the limit for a polynomial $p(x) = \sum_{n=0}^N a_nx^n$. For example, in the the specific linear case ...
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1answer
59 views

Finding a value that a set of given polynomials have the largest possible common divisor

Let $f_1, ..., f_n$ be homogenous polynomials in $Z[x_1, .., x_4]$. Find the value $\alpha=(\alpha_1, ... , \alpha_4)$ such that $f_1, ... , f_n$ evaluated at $\alpha$ have the largest common ...
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3answers
86 views

for which a, the matrix A is diagonalizable?

A = $ \begin{pmatrix} 2a+3 & 0 & 0 \\ -a-3 & a & a+3 \\ a & a & a+3 \\ \end{pmatrix} $ Characteristic polynomial: $ ...
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Derivation of composite Gaussian quadrature error formula

I am working on studying for the Numerical Analysis qualifying exams. One of the questions I am stuck on is the following: Derive the error term for the composite Gaussian quadrature rule with ...
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Irreducibility of some polynomial

Let $p(x) = (1+ \cdots +x^k)^2 + (1+ \cdots +x^k) + 1$, for some $k \geq 2$ fixed. I would like to know if $p(x)$ is irreducible in $\mathbb{Q}[x]$.
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2answers
87 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 ...
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0answers
39 views

How “separable” (not in that sense) is a polynomial?

Since "separable" is used for different meaning in separable polynomial and separation of variable, I am having trouble searching for anything related to my question. So I hope someone can help with ...
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3answers
42 views

In $\mathbb{Q}[x]$, what is the gcd of $x^6 − 1$ and $x^4 − 1$…Using Euclid's Method. [closed]

In $\mathbb{Q}[x]$, what is the gcd of $x^6 − 1$ and $x^4 − 1$...Using Euclid's Method. Could someone please do the first few steps of this so I know how to solve gcd of polynomials using Euclid's ...
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51 views

Representing a higher degree polynomial as product of smaller degree polynomials?

Consider an equation $H(Z)=1+\frac 52Z^{-1}+2Z^{-2}+2Z^{-3}$ I want to write it as a product of a first degree polynomial and another polynomial, which will be...$$H(Z)=(1+2Z^{-1})(1+\frac ...
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vector subspace of all real polynomials which are divisible with $x^2 + 1$

Show that the set of all real polynomials which are divisible with $x^2 + 1$ is a vector subspace of space of all real polynomials to 4th degree. Also find base and dimension of this subspace. I ...
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3answers
60 views

complex roots calulation question

How can we find the roots of an equation such as:$z^2 +z +1=0 ,z \in \mathbb{C} $ ?
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2answers
86 views

Find the roots of the equation $(1+xi)^n+(1-xi)^n=0$

Find the roots of the equation $f(x)=(1+xi)^n+(1-xi)^n=0$. I'm having problems finding the roots...this is what I've done: First I expressed $(1+xi)^n$ and $(1-xi)^n$ in trigonometric form and ...
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Determining if a given equation is solvable given a set of ultra-radicals

So suppose someone is armed with the tools of standard arithmetic, exponents (and of course that comes along with roots) AS WELL AS a set of inverses for some polynomials which are not solvable using ...
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1answer
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About $\mathbb{F}_7[x]$

can you help me with this? Let $a(x)=3x^6+2x^2+x+5$ and $b(x)=6x^4+x^3+2x+4$, find the g.c.d between $a(x)$ and $b(x)$ in $\mathbb{F}_7[x]$. Thanks!
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2answers
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Finding the remainder when a polynomial is divided by another polynomial. [duplicate]

Find the remainder when $x^{100}$ is divided by $x^2 - 3x + 2$. I tried solving it by first calculating the zeroes of $x^2 - 3x + 2$, which came out to be 1 and 2. So then, using the Remainder ...
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Finding a cubic polynomial whose zeroes are the same as collectively of two other quadratic polynomials.

The question is: Find a cubic polynomial $p(x)$ whose zeroes are the same as those collectively of polynomials $g(x) = 2x^2 - 9x + 4$ and $f(x) = 2x^2 + 3x - 2$. Given that $p(0)$ = 8. I tried ...
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1answer
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Finding the remainder polynomial for a given polynomial.

When a polynomial $p(x)$ of degree 3 is divided by $3x^2 − 8x + 5$, quotient and remainder obtained are linear polynomials such that $p(1)$ = 19 and $p(5/3)$ = 25. So, find the remainder polynomial. ...
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A bunch of questions involving polynomials.

Okay, so apparently, I can't write more than one post in the space of 20 minutes. So, I'm writing down all the questions I wasn't able to solve here. It would be great if you could solve them and ...
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Real polynomials, complex zeroes and the Intermediate value theorem

I have a second grad polynomial p(x). For arguments sake lets say $$p(x) = x^2 + 16x + 76$$ I also have an inequation $$p(x) > 0$$ Now the inequation does not have a real solution, but only ...
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2answers
46 views

Find the values of a,b and c in a polynomial $p(x) = ax^2 + bx + c$

The question is this : A polynomial $p(x) = ax^2 + bx + c$ where $a,b,c$ are some rational numbers, has $1 + \sqrt3$ as one of the zeroes and also $p(2) = -2$. Find the values of $a,b$ and $c$. ...
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Find the value of “k” so that the quadratic polynomial has equal zeroes.

The question is this: Find the the value(s) of $k$ so that the quadratic polynomial $kx^2 + x + k$ has equal zeroes. Answers along with appropriate explanations would be appreciated. Thanks.
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3answers
516 views

What is the lowest-degree function that passes through these points?

I want to find a (preferably polynomial) function that passes through the following twelve points: $(1, 0)$ $(2, 3)$ $(3, 3)$ $(4, 6)$ $(5, 1)$ $(6, 4)$ $(7, 6)$ $(8, 2)$ $(9, 5)$ $(10, 0)$ $(11, ...
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2answers
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Build field extension and solve equation

Build quadratic extension of field that contains $5$ elements. And solve $x^2+x+2=0$ in this field. As I understand we need to build $\mathbb{F}_{5^{2}}$. Field $\mathbb{F}_5$ contains ...
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2answers
60 views

Matrix with rank 3 does not exist in this $p(x)$

Given: Characteristic polynomial is $p(x) = x^7 - x^5 + x^3$ . Prove that there isn't a matrix A that $ \rho(A) = 3 $ I tried to play with $p(x) = x^3(x^4 - x^2 +1)$ But I'm still not sure how ...
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1answer
42 views

Polynomials - getting wrong answer using Euclidean algorithm

I am finding the GCD of $a = x^3 + 11/3x^2 + 17/4x + 3/2$ and $b = 3x^2 + 22/3x + 17/4$ using the Euclidean algorithm. So I divide $a/b$ and get $q$ and $r$ such that $a = qb + r$. Then, according to ...
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Need help with a diophantine expression

I'm faced with this problem. Under what conditions is this expression a positive odd integer: $$\frac{2^g(x^2+y^2-z^2)}{x+y-z}$$ where $g,x,y,z$ are nonnegative integers. x and z are odd, and y is ...
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1answer
19 views

Descartes rule of signs extension

Let $V(\text{sequence})$ be the number of sign changes in the sequence, e.g. $V(-3,0,-2,9,0,1)=1$. Show that $V(a_0,a_1,...,a_n)\ge V(a_0,a_0+a_1,a_0+a_1+a_2,...)$. Furthermore, prove that if ...
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3answers
54 views

A closed formulae for the coefficient of $x^k$ in $(x-1)^a(x+1)^b$

Let a,b positive integer Do you know any closed formulae for the coefficient of $x^k$ in $(x-1)^a(x+1)^b=\sum_{k=0}^{a+b}u(k;a,b)x^k$ ? I look for an a closed expression of $u(k;a,b)$ involving ...
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0answers
111 views

Reduced Gröbner basis and extension of scalars

Consider a field extension $L\subseteq K$, and let $\mathfrak a\neq 0$ be an ideal of the polynomial ring $L[T_1,\ldots,T_n]$. Suppose that a monomial order is fixed, so there exists a unique reduced ...