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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Factorization Process in a polynomial ring

Reading the book "Field Theory" by S. Roman, in chapter $0$ I found the following problem: Let $F$ be a field and consider the polynomial ring $F[x_1,x_2,\ldots]$ where $x_i^2 = x_{i-1}$. Show that ...
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2answers
57 views

Find a polynomial with cubic values for consecutive integers.

Can we find a cubic polynomial (except the obvious ones, i.e. cubes of linear polynomials), say, $f(x)\in \Bbb{Q[x]}$ whose values are cubes for four consecutive integers? What about five consecutive ...
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1answer
61 views

A question about polynomial in two variables [duplicate]

Let $f:\mathbb R^2 \to \mathbb R$ be a function such that for any $b \in \mathbb R$ , the function $f_b : \mathbb R \to \mathbb R$ defined as $f_b(x):=f(x,b) , \forall x \in \mathbb R$ , is a ...
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0answers
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$N$-dimensional linear operator is normal, Lagrange interpolation?

Is there a way to see that an $N$-dimensional linear operator $A$ is normal if and only if $A^\dagger$ can be represented as a linear combination of $I, A, A^2, \dots, A^{N-1}$ using Lagrange ...
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1answer
25 views

Find all the values of $k$, if any, such that $f=t^4+2t^3-3t^2+2kt+k^2$ is divisible by $g=t+2$ in $\mathbb{Z}_{7}[t]$

Find all the values of $k$, if any, such that $f=t^4+2t^3-3t^2+2kt+k^2$ is divisible by $g=t+2$ in $\mathbb{Z}_{7}[t]$. I solve it in the normal way but I do not sure that my way is correct or ...
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43 views

Prove $f(x)$ has no rational number roots

Let $f(x)=ax^2+bx+c, \ a,b,c\in \mathbb{Z}$, and such $$|f(1)|,|f(2)|,|f(3)|,|f(4)|,|f(5)|$$ are prime numbers, show that $f(x)=0$ has no rational number roots
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Optimal step for drawing Bezier curve

Bezier curves are parametric in the sense that for each dimension their polynomials share common parameter $t$ [1]. To draw a Bezier curve on screen one could increment $t$ by tiny step and calculate ...
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1answer
48 views

Example of a diophantine polynomial

A diophantine set is a subset of a power $\mathbb{Z}^k$ of the set $\mathbb{Z}$ of integers which can be written as $$\{x \in \mathbb{Z}^k : \exists y \in \mathbb{Z}^m : P(x, y)=0\}$$ where $P$ is a ...
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63 views

Polynomial with exactly one complex root

Is it possible that a polynomial of degree $n$ with real coefficients has exactly one complex root? I saw https://en.wikipedia.org/wiki/Complex_conjugate_root_theorem but wondered if this can happen ...
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About $(x^3 - 4)^2 - x^6 + 2x^5 = 2x^5 -8x^3 + 16$

Studying polynomials I got the follows: $$ (x^3 - 4)^2 - x^6 + 2x^5 = 2x^5 -8x^3 + 16 $$ I can't understand from where we got this $-8x^3$. I got to simplify this polynomial just to: $$ 2x^5 + 16 ...
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Show that $x^3-3$ is irreducible in $\Bbb Z_7[x]$.

Show that $x^3-3$ is irreducible in $\Bbb Z_7[x]$. In the text, we haven't gotten to the theorem that the roots of polynomials are the only factors , and I would rather not prove it in this ...
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0answers
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Find Zeros / Factors of a polynomial

I have been told that to find factors of a polynomial (nth degree) we have to find the factors of constant term and that of coefficient of leading term of the polynomial in concern. The possible ...
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389 views
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A polynomial sequence

I have a sequence of polynomials $Q_k(x, y)$, $k\geq 1$ defined recursively as follows: $Q_1=x$. There is a sequence of polynomials $p_j(y)$ of degree $j$ such that $Q_{2m}$ is of the form ...
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3answers
45 views

Factoring Trick - Adding Up Coefficients

My professor told me this for factoring polynomials: Add up the coefficients and if they equal 0 then the polynomial has root of 1. Add up, but switch the signs of the coefficients with odd ...
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1answer
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If $f,g$ in $Z[x]$, $h$ in $R[x]$ with $f=gh$, is $h$ nessecarily in $Z[x]$?

Let $f$ and $g$ be monic polynomials in $Z[x]$. There exists a polynomial $h$ in $R[x]$ such that $f=gh$ for all real $x$. Is $h$ nessecarily in $Z[x]$?
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Is an analytic one-to-one function on the whole plane necessarily a polynomial? (Can it be disproved?)

I had to show what a one-to-one analytic function from the plane to itself could possibly be. So, I studied the behavior of such a function at infinity: Case 1: Such a function cannot have no ...
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31 views

How to find a perfect regression fit in R?

I have a set of points, which I know can be described with some equation. How can I find this equation? The scatter plot for this set looks like this: I look at the plot and assume that I can use a ...
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4answers
61 views

When are we permitted to multiply or divide both sides of an equation by a variable?

As it is said in the mathematics books (at least the one I have), we are not permitted to divide or multiply both sides of an equation by a variable, because it is possible to lose some answers. For ...
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20 views

Polynomial interpolation- numerical analysis

Using polynomial interpolation show that the polynomials $p(x) = (x-1)(x-2)(x+1) $and $f(x) = x^3 - 2x^2 - x + 2$ are one and the same......its absurdly simple...... I tried to show by Newtonian ...
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43 views

How to confirm the number of real root of a quartic equation? [closed]

Given that I have a quartic equation as follow $$a x^4 +b x^3+c x^2+d x +e=0$$ Now I want to know the number of real root. I search something by...
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37 views

Factorial-Like Symbol for polynomials?

Is there a symbol similar to the factorial for polynomials? Like if I say $4!=4\times 3\times 2\times 1$ What is the equivalent operation such that Operation: $(x)(4)=x^4+x^3+x^2+x$ Where $x$ is ...
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1answer
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intersection of maximal ideals in a polynomial ring

Given $A=K[x_1,\dots,x_n]$ a polynomial ring on a field $K$, let $p(x)\in A$ be an element, and $M_1,\dots,M_s$ some maximal ideals. Is it true that $$\cap(M_i,p) = (\cap M_i,p)?$$ I obtained that ...
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You are given a series of random points and asked to find a graph that fits them, how would you do it?

If you were given a large series of random number pairs $(x,y)$ and asked to find a graph that fit all of them in the form of $y = Ax^Z + Bx^Y + Cx^W + \cdots $. Fine by me if sins and coses and ...
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3answers
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Find coefficients of polynomials $f(x)=x^2+ax+b$ and $g(x)=x^2+cx+d$ $(a,b,c,d \in \mathbb{R})$

Roots of polynomial $f(x)=x^2+ax+b$ are cubes of the roots of polynomial $g(x)=x^2+cx+d$. Sum and product of roots of polynomial $g(x)$ are equal. Find coefficients $a,b,c,d$ so that polynomial $f(x)$ ...
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Explicit Linear Matrix Inequality representation of sum of square cones.

The cone of sum of square (sos) polynomials is a projected spetrahedron. This should means we can find a Linear Matrix Inequality (LMI) whose projection is sos cone. For example, the quartic sos ...
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1answer
27 views

Why doesn't $f_1, f_2$ have an inverse polynomial?

Consider $f\in R=F[X]$. It is given that $f$ doesn't have an inverse but it's reducible. Therefore, there are $f_1,f_2$ such that $f=f_1f_2$, where $f_1, f_2$ also doesn't have an inverse polynomial. ...
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Why is there a $q_i$ such that $q_j|q_i$?

Let $q_i$, a sequence of of irreducible polynomials where $q_i$'s highest-order term has coefficient $c_n = 1$ (by the way, what's the right term to describe this property?) Anyhow, let's look at: ...
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1answer
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A problem in Algebra [duplicate]

If $a+b+c+d=0=a^7+b^7+c^7+d^7$, Then prove $a(a+b)(a+c)(a+d)=0$ and all are real numbers. I am confused and don't even know which to tag for help.
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What kinds of functions have fixed points?

Among continuous functions, can we characterize those which have fixed points and those which do not? Geometrically, these are the functions that intersect the line $f(x) = x$. Is that the most ...
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C++: Finding roots of a polynomial ring.

I need a library (can be used in C++) that finds the roots of polynomial over $\mathbb{Z}_p$ where $p$ is a large prime number. So I can give it an array of coefficients that are big integers in the ...
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Second Degree Equations

I am having problems figuring out how to solve the following second degree equations: 2x$^2$ + 3x + 1 = 0 I can't get factors that add together to get 3 or multiply together to get 1: (2x + ?)(x ...
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finding all finite fields making $x^2+x+1$ irreducible [duplicate]

Find all finite fields making $f (x)=x^2+x+1$ irreducible. Obviously, we have only two cases: For each prime $p $, $f $ is reducible over $\mathbb {F}_p $. Then, for any $n\ge 1$, it's reducible ...
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Showing that the roots of the quadratic are real

If $x^2+bx+c=0$ has real roots, show that the roots of the equation $x^2+bx+c(x+a)(2x+b)=0$ are real for all real values of $a$. I could do it by standard way by proving determinant is postive. ...
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2answers
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How to use the factor theorem on $a(b^2-c^2) + b(c^2-a^2) + c(a^2-b^2)$?

I know the factor theorem i.e, Let $P(x)$ be a polynomial of degree greater than or equal to $1$ and $a$ be a real number such that $P(a) = 0$, then $(x-a)$ is a factor of $P(x)$. I have an question ...
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1answer
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Sequence of polynomials converging to a discontinuous function in $\mathbb{C}$ [closed]

Is there a sequence of polynomials $P_n $ such that $\displaystyle\lim_{n\rightarrow\infty} P_n (z) $ exists everywhere in $\mathbb{C}$ and equals to $1$ if $\text{Im} (z)>0$, $0$ if $\text{Im} (z) ...
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Difference between a polynomial of degree $n$ and an $n$- tuple with the $n$th component $\neq 0$?

When a polynomial $f := a_0 + a_1 X + \cdots + a_n X^n$ over a field with $a_n \neq 0$ cannot be regarded as a function, what is the major difference between $f$ and the $n$-tuple $(a_0,\dots, a_n)$? ...
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Ruffini's Rule with parametric binomial

I'm having some problem in applying Ruffini's Rule in the case the binomial contains a parameter. The problem is: $$[x^4-(a-3)^2x^2+ax+3a+a^2]:(x+a-3)$$ I'm assuming $$ \begin{array}{c|cccc|c} ...
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Two variable quadtraic polynomials geometric representation?

We have learned about quadratic polynomials having two variables but I ran into the question of do all of these type of polynomials have empty sets? So... Given a quadratic polynomial in two ...
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Computing as many digits as possible of $\sqrt{2}$ with a pen and a paper in 5 minutes

You have to compute as many digits as possible of $\sqrt{2}$ with a pen and a paper (an eraser if you're lucky...) in 5 minutes. What will you do? What is your justification for doing it? The ...
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1answer
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Trinomial expansion variation - generalize?

One may represent $(1+x+x^2)^k$ = $\sum_{\ell=0}^{2k}\begin{pmatrix}k\\l-k\end{pmatrix}_2x^\ell$, where $()_2$ is the trinomial coefficient. Any one with experience how to represent $(1+x+0.5x^2)^k$ ...
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Is this induction proof of the Fundamental Theorem of Algebra rigorous?

I was trying to find a suitable proof of the Fundamental Theorem of Algebra at an undergraduate level which avoided abstract linear algebra, as I have not yet begun it. However, I came across this ...
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Bernstein polynomial

I need some help in the following task. The i-th Bernstein polynomial of degree n on the interval [a,b] is $B_{i}^{n}(x;a,b) = (b-a)^{-n}\binom{n}{i}(b-x)^{n-i}(x-a)^{i}$ Show: The control points of ...
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1answer
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Nature of roots of a biquadratic equation

(Biquadratic $\rightarrow$ Quartic (degree 4)) The Question: (from a book i am practicing from) Find the nature of the roots of the equation $$f(x) = 45 x^4-144 x^3+146 x^2-56 x+12=0$$ (By nature i ...
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Deriving the formula for the Quartic

Could someone explain to me how was the Quartic formula originally derivated? Is there any simple 'modern' method for deriving it? How does Galois theory help?
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Can the roots of $f(x)=x^4-x^3+2x^2-x-1$ be found algebraically?

Can the roots of $f(x)=x^4-x^3+2x^2-x-1$ be found algebraically? Are there multiple methods for doing so?
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Multivariate “base polynomials”

Fix a positive integer $b$. For any positive integer $N$ whose base-$b$ expansion is $\sum \alpha_k b^k$, define $p_N(x)$ to be the polynomial $\sum\alpha_k x^k$. Goal: Produce a polynomial ...
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1answer
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Quadrature formula on triangle

I am looking for a quadrature formula on the triangle, with points at the vertices and at the mid-edges, so 6 points, and that is exact for polynomials of degree at least 2, with weights strictly ...
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2answers
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STEP past question: Showing that $a^2$ is a root of the following equation

I'm having difficulty with the following question The first part seems simple enough, and by expanding the right hand side I get that $p=-a^2+b+c$ $q=a(b-c)$ $r=bc$ But when asked to show that ...
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2answers
122 views

$f\cdot g=0 \implies f=0 $ or $g=0$.

I know this is kind of an obvious thing to say: Let $f,g \in \Bbb K[x]$, then $$f\cdot g=0 \implies f=0 \text{ or } g=0$$ But to my surprise I couldn't prove it. What's a simple way to do this?
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1answer
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Partial derivative of polinomial root

I have a characteristic equation of the form $P(x,y,z) = 0$. $P$ is a polynomial in $x$ with degree of 3 and is a first order polynomial in $z$. I computed the value of $x=F(z)$, such that ...