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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Polynomial factors

Why must $x^2 + x + 1$ be a factor of $x^5+x^4+x^3+x^2+x+1$? I know that when we divide $x^5+x^4+x^3+x^2+x+1$ by $x^3+1$ we get $x^2 + x + 1$, but is there an argument/theorem or anything that ...
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A system of polynomial-like equations

Let $p\in \left[0,1\right]$ and take $a_1,a_2,\ldots,a_n\in \mathbb{R}^{+}$. What is the maximum number of solutions that the system of (nonlinear) equations $$x_1^p +x_2^p +\cdots+x_n^p = 1\\ ...
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What is the galois group of $x+3$ or $(x+1)(x+2)$ ? How about $A(x)B(x)$?

As the title says I wonder what the galois group of $x+3$ is. Or even if that exists ? Since $x+3 = 0$ has only one zero/element I assume its the trivial group ? And what is the galois group of ...
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Automorphisms of $\mathbb{F}_2[x,y]$

What are the automorphisms of the 2-variable polynomial ring over $\mathbb{F}_2$, the field with 2 elements? Are they generated by $(x \mapsto y, y \mapsto x)$, $(x\mapsto x+ p(y), y\mapsto y)$, and ...
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122 views

Factoring polynomials of degree $a p^b$ over extension fields.

Let $f(x)$ be an irreducible polynomial with integer coefficients, which is irreducible over $\mathbb{Z}$ and has degree $a p^b > p$ with $p,a,b>0$ and $p$ a prime. It appears that $f(x)$ ...
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How to tell that $x^2 - x \sin(x) -\cos (x)$ has no real roots?

Putting the equation $x^2 - x \sin(x) - \cos (x)$ into Wolfram Alpha, I am surprised that it has a nice parabolic shape. Also, it has two complex roots. Question Is it possible to tell, in a simple ...
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Discriminant and roots of $ x^{n^2} \pm (x-1)^{n^2}$?

When considering the polynomials $x^{n^2} \pm (x-1)^{n^2}$ ( $n$ integer > 1 ) i noticed some things that appeared weird to me. Discriminant($x^{n^2} + (x-1)^{n^2}) = (n^2)^{n^2}$. ...
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898 views

prove that the entire function f is a polynomial.

Suppose that $f$ is an entire function, and that in every power series $f(z)=\sum_{n=0}^{\infty} c_{n}(z-a)^n$ at least one coefficient is 0. Prove that $f$ is a polnomial. Hint: ...
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159 views

How to find the roots of this polynomial

The polynomial: $8x^4-8x^2+1=\frac{\sqrt{3}}{2}$ I can simplify with $u=x^{2}$ to $8u^2-8u+{\frac{\sqrt{3}}{2}}=0$ Mistake $\left(1-\frac{\sqrt{3}}{2}\right)$ apply the quadratic formula: ...
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Polynomial Formula like Infinite Sum with non-natural index

By polynomial formula $$(\sum_{i\in m} x_i)^n=\sum_{\substack{j_i \in \mathbb{Z}^+ \\ \sum j_i=n}}\left(\begin{array}{c} n\\ j_{0},\ldots , j_{m-1} \end{array}\right)\prod_{i \in m} x_i^{j_i}$$ where ...
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$(1-t)^{-d}= \Sigma_{k=0}^\infty {d+k-1 \choose d-1} t^k$?

I'm trying to see why the equation $(1-t)^{-d}= \Sigma_{k=0}^\infty {d+k-1 \choose d-1} t^k$ holds in the power series ring $\mathbb{Z}[[t]]$. I assume it's a counting argument about the number of ...
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bound the distance of two roots of multivariate polynomial systems

Consider a system of multivariate polynomial equations $\vec{x}= f(\vec{x})$ with integer coefficients, $f$ is at most of degree 2. Suppose $\vec{x}_1$ and $\vec{x}_2$ are two real roots of $f$, is ...
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477 views

Zero-divisors and units in $\mathbb Z_4[x]$

Consider the ring $\mathbb Z_4[x]$. Clearly the elements of the form $2f(x)$ are zero divisors. 1. Is it true that they are all the zero divisors? I mean is it true that if $p(x)$ is a zero ...
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72 views

Estimation of a polynomial

I'm currently reading the following paper: http://arxiv.org/abs/1209.0612 and got stuck on Proposition 3.1 (2). The claim translated to polynomials is the following: Assume $n\geq 3, c\geq 1, ...
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polynomials over a local Artinian (or finite) ring

In this question " Zero-divisors and units in $\mathbb Z_4[x]$ " it looks like it has been shown that the set of zero divisors of $\mathbb{Z}_4[x]$ coincides with its nilpotent elements. Since the ...
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149 views

Three inequalities with sums of fractions over two positive integers

In a proof, I arrive at three inequalities for all $p,q \geqslant 0$: \begin{align} \frac{p+1}{q+1} + \frac{q+1}{p+1} &\geqslant 1 + \frac{p}{2q+1} + \frac{q}{2p+1} + \frac{1}{p+q+1};\cr ...
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386 views

Quotient ring of a polynomial ideal with two variables

Given an ideal $I = \langle x-y,y^3+y+1 \rangle \subset \mathbb{C}[x,y]$ (this is a Gröbner basis w.r.t. degree-lexicographic order). I want to write $\mathbb{C}[x,y]/I$ as a $\mathbb{C}$-Basis and ...
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387 views

How to define a bijection between natural numbers and the set of all polynomials with natural coefficients and finite variables?

Is there an explicit algorithm which establish a bijection between polynomials with finite variables and natural coefficients and natural numbers. Does anyone have one of these?Thanks.
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Does the sign of the characteristic polynomial have any meaning?

The characteristic polynomial of a matrix $A \in \mathbb{C}^{n \times n}$, $p_A (\lambda) = \det(A-\lambda \cdot E)$ can be used to find the eigenvalues of the linear function $\varphi:\mathbb{C}^n ...
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53 views

Two bivariate rational functions

I have $2$ bivariate rational equations: $\dfrac{t(b_2^2-a_2^2-t^2b_2^2)+t^3a_2^2}{1+t^2}+\dfrac{ta_1a_2-ts^2a_1a_2-b_1b_2s+b_1b_2st^2}{1+s^2}=0$ ...
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A convex polynomial inequality.

Let $\phi(x)$ be a convex polynomial of degree $m$ at least two. Note that for $x,q \in \mathbb{R}$ $$\phi(x) + \phi(q) - 2\phi(\frac{x+q}{2}) = ...
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77 views

$y=ax^2+bx+c$, $a,b,c \in \mathbb{R}$, $a > 0$ show that $c \ge -\frac{1}{4a}$

I am having trouble solving this. I know that the vertex is $\left(-\frac{b}{2a}, p(-\frac{b}{2a})\right)$, where $p(x) = ax^2+bx+c$, which is $(-\frac{b}{2a},c)$. After that I am lost, how to show ...
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Positive definite completion of a matrix

Suppose we have a real, symmetric matrix $A(x_1,x_2,x_3)$ given by \begin{pmatrix} a_{1,1} & a_{1,2} & x_1 & x_2 \\ a_{2,1} & a_{2,2} & a_{2,3} & x_3 \\ x_1 & a_{3,2} & ...
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290 views

Obtaining the discriminant of the characteristic polynomial directly from the matrix

Let $M \in \mathbb{Z}_{n \times n}$ be a square matrix with integer coefficients. Let $P(x)$ be its characteristic polynomial $$ P(x) = \det\left(x \cdot \mathbb{I}_{n \times n}- M\right) $$ I ...
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Graphing a line of an expected value

Given E[x]=3, var[x]=9, Graph the line y=(x-1)(x-2)(x-3). How does one graph such a thing?
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Dimension of the splitting field of $f = x^3 − x + 1$

Let $L_f$ be the splitting field of the irreduicble polynomial $f = x^3 − x + 1$ over $\Bbb{Q}[x]$. I want to determine $\operatorname{dim}_{\Bbb{Q}}L_f$. $f$ has three roots in its splitting field ...
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Splitting field and dimension of irreducible polynomials

Given a field extension $L/K$, $\alpha, \beta \in L$ and $f,g \in K[x]$ irreducible polynomials with $f(\alpha)=g(\beta)=0$. Then $$ \operatorname{dim}_K(K(\alpha,\beta)) = \deg(f) \cdot ...
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Why $x^{p^n}-x+1$ is irreducible in ${\mathbb{F}_p}$ only when $n=1$ or $n=p=2$

I have a question, I think it concerns with field theory. Why the polynomial $$x^{p^n}-x+1$$ is irreducible in ${\mathbb{F}_p}$ only when $n=1$ or $n=p=2$? Thanks in advance. It bothers me for ...
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Dimension of a splitting field

Given a field $F$ and a polynomial $P \in F[x]$ such that $P$ is irreducible over $F$. Let $L_P$ be the splitting field of $P$ and $F$. Does $\operatorname{dim}_F({L_F}) = \deg(P)$ hold? If looking ...
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Alternative solution to determine the number of irreducible, monic polynomials in $\mathbb{Z}_p[x]$ of degree $k$

I know the problem of the number monic, irreducible polynomials of degree $k$ in $\mathbb{F}_p$ have been discussed and that there is a general formula which solves this problem. Nevertheless, I have ...
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143 views

Irreducible polynomials on $\mathbb Z_2$ and $\mathbb Z_k$

I had to solve many exercises to take my algebra exam lately and lots of them are on polynomials, whereas I am asked to look for roots and verify if the given polynomials are reducible in a specific ...
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68 views

Distinct-degree factorization in finite fields

In $\mathbb{Z}_3$ $$ x^9 : x^4+x^3+x^2+2x+1 = x^5+2x^4+2x^2+2x$$ with remainder of $x$. In $\mathbb{Z}_7$ $$x^7 : x^4+5x^3+x+5 = x^3+2x^2+4x$$ with remainder of $x$. Is this random? Or is there ...
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Finding a polynomial $g$ such that $g^2=f$ for certain $f$ in $\mathbb{F}_{16}$

Let $L = \mathbb{Z}_2[x]/\langle x^4 + x + 1 \rangle$ and $\alpha := [x] \in L$. I want to find $g \in L[y]$ with $g^2 = f$ and $$f = (\alpha^2 + \alpha)y^8 + \alpha y^4 + \alpha^3y^2 + \alpha + 1 ...
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92 views

Is it possible to extend phi to higher order polynomials?

Below, $x=\phi$ when $n=2$: $$x^n-\sum_{i=1}^{n}x^{n-i}=0$$ ($\phi$ being the golden ratio) Is there a way to express $x$ in terms of $\phi$ for $n>2$?
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151 views

Minimal polynomial of $i\frac{\sqrt{3}}{2}+\frac{1}{2}$

I'm looking for the minimal polynomial of $\alpha = i\frac{\sqrt{3}}{2}+\frac{1}{2}$ in $\mathbb{Q}[x]$. A polynomial with a root $\alpha$ is $$(2(x-\frac{1}{2}))^4-9.$$ A computer algebra system ...
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271 views

solving for a coefficent term of factored polynomial.

Given: the coefficent of $x^2$ in the expansion $(1+2x+ax^2)(2-x)^6$ is 48, find the value of the constant a. I expanded it and got ...
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Transforming root-equations into polynomials

Let's define special polynomials as polynomials in $\mathbb{Q}[X]$, where we allow to make roots, too. Examples: $\sqrt{X^4+1}$, $\sqrt[3]{X}+\sqrt{X+1}$, $\sqrt{X+\sqrt{X+1}}$ How can I transform a ...
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Small math help with polynomials

If one solution of the equation $3x^2 = 8x + 2k + 1$ is $7$ times the other. Find the solutions and the value of $K$. Note: This isn't a homework question. I'm skipping ahead in my textbook. Thank ...
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Easy polynomials question?

Please do this without using the quadratic formula. If $\alpha$ and $\beta$ are zeroes of the polynomial $x^2 -6x + a$ then find the value of "$a$" if $3\times \alpha + 2\times \beta = 20$ Thank you ...
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factor theorem with two variables

is it possible to use the factor theorem when there is more than one variable? I believe so; however, don't know how to check every case. Example: $x^2-y^2$
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403 views

Factor $4x^3-8x^2-25x+50$ completely

Factor $4x^3-8x^2-25x+50$ completely The highest numbers you can take would be $1$, $2$, or $4$. Neither of those apply to all. So let's try the $x$! But the last term $50$ doesn't have an $x$ ...
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Reduced Gröbner Basis

If have computed this Gröbner Basis with Buchberger's algorithm for Degree-Lexicographic-Ordering: $$\{ x^²y+x+1,xy^2+y+1,x-y \} $$ I want to to transform it into a unique representation form called ...
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How to prove $t^{23}+1$ irreducible in $F_p$?

I have tried to prove that $t^2+1$ is irreducible over $F_3$ by supposing to the contrary $t^2+1=(t+\alpha)(t+\beta)=t^2+(\alpha+\beta)t+\alpha\beta$. Then, $\alpha+\beta\equiv 0 \pmod 3, ...
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Why $t(x)$ is a divisor for every $\gcd(h(x),k(x))$

Determine two polynomials $h(x), k(x) \in \mathbb Q[x]$ having rispectively $1,-1,2$ and $1,2,-2$ as roots. Explain why $t(x) = x-1$ is a divisor for every $\gcd(h(x),k(x))$. I figured $h(x) ...
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Does the Abel-Ruffini Theorem contradict the Fundamental Theorem of Algebra?

It is my understanding that the Abel-Ruffini Theorem implies that certain polynomial equations $(x^5-x+1=0$, for instance) have transcendental roots. However, the Fundamental Theorem of Algebra states ...
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2answers
137 views

Solving polynomials in $\mathbb{Q}[X]$ exactly

I wanted to write an equation solver for rational polynomials in one variable $X$. However, such solutions do not need to be in $\mathbb{Q}$. What I wanted was to display solutions "lossless", i.e. ...
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1answer
96 views

Describing the ideals for which $\operatorname{dim}_F(F[x,y)]/I) = 4$

Given a field $F$. I want to describe the ideals $I$ such that $\operatorname{dim}_F(F[x,y]/I) = 4$ (with Groebner Basis). I have an understanding of Groebner Basis but I lack an intuition of what the ...
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75 views

Help to show if the function is decreasing for large $l$

I would like to see if $$ b_l:=4^{-l} \sum_{j=0}^l \frac{\binom{2 l}{2 j} \binom{n}{j}^2}{\binom{2 n}{2 j}}\text{.} $$ is decreasing when $l$ is large enough say around $10^6$. I dont need any ...
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45 views

Calculating the coefficients of one form of bi-variate polynomial from its another form

Given a polynomial: $$ P(x_1, x_2) = (ax_1+b)(cx_2+d)$$ This can be written in another form as: $$ P(x_1, x_2) = d_1x_1x_2 + d_2x_1 + d_3x_2 + d_4$$ where, $d_1 = ac$, $d_2 = ad$, $d_3 = bc$, $d_4 = ...
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Cubic with complex roots

I have a problem figuring out how exactly I find the cube roots of a cubic with complex numbers. I need solve the cubic equation $z^3 − 3z − 1 = 0$. I've come so far as to calculate the two complex ...