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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2
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1answer
101 views

Coefficient signs in the sum of successive powers of a polynomial

I'm searching for some structure in the sign variation of the coefficients of: $$P = \sum_{i>0} p^i\enspace,$$ for some polynomial $p \in \mathbb{Z}\langle x\rangle$ with no constant term. I'm ...
1
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1answer
75 views

Is the ideal $(X_0X_1+X_2X_3+\ldots+X_{n-1}X_n)$ prime?

Consider the ideal $(f = X_0X_1+X_2X_3+\ldots+X_{n-1}X_n)$ in the polynomial ring $k[X_0,\ldots, X_n]$. Is this a prime ideal? If so, what is its height? I'm stuck trying to show that $f$ is ...
6
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4answers
2k views

Help with Cardano's Formula

I'm trying to understand how to solve cubic equations using Cardano's formula. To test the method, I expand $(x-3)(x+1)(x+2)=x^3-7x-6$. My hope is that the formula will produce the roots $-1,-2,3$. ...
2
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0answers
104 views

Generalizing an approach to proving AMGM

This problem is Exercise 5.5.30 of "The Art and Craft of Problem Solving" by Paul Zeitz. The problem asks to use the identity $$ a^3+b^3+c^3-3abc = (a+b+c)(a^2+b^2+c^2-ab-ac-bc) $$ to prove the AMGM ...
1
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1answer
59 views

overdertermined system of polynomial equations

From linear algebra i understood that if the system of linear equations are independent of each other and if the number of equations is more than the number of variable, then the system is ...
3
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1answer
236 views

Generating set for sum of two ideals

Suppose there are two ideals $I,J \in \mathbb{C}[x_1,\dots,x_k]$ and two sets of generating polynomials $\langle f_1, \dots, f_s\rangle$, $\langle g_1, \dots, g_t\rangle$. Now I want to describe $I + ...
2
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0answers
218 views

$L_2$-norm representation of the function

Let $$ f^{\alpha}_+(x)=\frac{1}{\Gamma(\alpha+1)}\sum_{k\ge 0}(-1)^k{\alpha+1 \choose k}(x-k)^{\alpha}_+, $$ where $\alpha > -\frac 12$(see for reference ...
2
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2answers
100 views

Solutions to $z^3 - (b+6) z^2 + 8 b^2 z - 7+b^2 = 0, b\in \mathbb R, z \in \mathbb C$

$z_1 = 1+i$ is a given solution. I guess what I have to find is $z_2$ and $z_3$ in $(z - (1 + i))(z - z_2)(z-z_3) = z^3 - (b+6) z^2 + 8 b^2 z - 7+b^2$. I tried to divide the polynomial by $(z - (1 ...
0
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1answer
114 views

Degrees of polynomials?

Let $A\left(x\right)$ represent a polynomial with a degree of $n-1$. Split $A\left(x\right)$ into odd and even powers. For example: $A\left(x\right) = 3 + 4x+6x^2+2x^3+x^4+10x^5$ $= \left(3+6x^2 + ...
4
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1answer
200 views

Polynomial inequality

I found the following problem on a website and would be curious to find a solution. Let $a_1\ge a_2\ge\cdots\ge a_n$ be real numbers such that for all integer $k>0$: $$a_1^k+a_2^k+\cdots+a_n^k\ge ...
3
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2answers
232 views

How to multiply two polynomials represented by values at distinct points?

I have two polynomials of degree $d$. However, I do not have equations for them. I simply have $d + 1$ distinct points on each polynomial. How would I find the product of these polynomials without ...
2
votes
1answer
284 views

How is a degree-$d$ polynomial uniquely characterized by its values at $d+1$ distinct points?

A degree-$d$ polynomial is uniquely characterized by its values at any $d+1$ distinct points. Could someone explain why the statement above is necessarily true?
4
votes
1answer
133 views

When does an irreducible polynomial stay irreducible in a cyclotomic extension?

Suppose that $P(x)\in\mathbb{Q}[x]$ is irreducible over $\mathbb{Q}$, and let $K$ be the $n$-th cyclotomic field. Is there a simple criterion to tell if $P$ remains irreducible over $K$? (Preferably ...
3
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2answers
2k views

Eigenvalues of a matrix and its square

Ok so I messed up my last question, I'll rephrase it: Is there a matrix $A$ of real elements, for which this holds true: $A^2$ has more unique eigenvalues than $A$. If not, then what about if the ...
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2answers
82 views

Characteristic polynomials of matrices

Good day! Given a characteristic polynomial $P$ of matrix $A$ I need to show that the characteristic polynomial $O$ of $A^2$ can't have more different real roots than $P$. I know that the ...
4
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3answers
616 views

The coefficient of $x^{18}$ in $(1+x^5+x^7)^{20}$

I was asked about a simple question that is: "What is the coefficient of $x^{18}$ in $(1+x^5+x^7)^{20}$? Generally, we know that; $$(x+y+z)^n= ...
5
votes
1answer
273 views

Antiderivative of Polynomials

I really like how differentiation is introduced for polynomials: Let $P(t) \in A[t]$ : $$D_P(t,s) = \frac{P(t) - P(s)}{t-s} \;\; \in A[t,s]$$ and the derivative of $P$ is $$P'(t) = D_P(t,t).$$ It ...
2
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0answers
88 views

What does it mean for a subset of $\mathbb{C}[x_1,\dots,x_n]$ to be algebraically independent?

What does it mean for a subset of $\mathbb{C}[x_1,\dots,x_n]$ to be algebraically independent? Particularly I'd like to know the formulation thereof which concerns the kernel of a surjective ring ...
1
vote
1answer
315 views

Linear interpolation for finding root of $f(x)$

When using linear interpolation, with similar triangles, to find the root of a function you narrow down the interval the root is in. If $f(1) < 0$ and $f(2) > 0$ then the root is in $[1, 2]$ ...
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3answers
708 views

Interval bisection to find a root of f(x)

I'm attempting to understand Interval bisection. I'm given a simple question in my textbook, and I can do the process easily, I just don't know when to stop. The question is "Use Interval bisection to ...
4
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1answer
158 views

Is the function “signomial”?

Function $f:(0, \infty)\longrightarrow \mathbb{R}$ is called $\textbf{signomial}$, if $$ f(x)=a_0x^{r_0}+a_1x^{r_1}+\ldots+a_kx^{r_k}, $$ where $k \in \mathbb{N}^*:=\{0,1,2, \ldots\}$, and $a_i, r_i ...
2
votes
3answers
169 views

When is this polynomial equal to a square?

When is $f(k):=8k^2+8k+1$ a square for $k\in\mathbb Z_{\geq 0}$? How do I begin on this? I see $f(k)$ is a square for $k=0,2$, but I do not know where to go from here.
5
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1answer
96 views

Agreement of two polynomials

How can I prove that if two polynomials (of matrix coefficients) agree in a neighborhood of $0$, then they are identical? Thank you!
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1answer
165 views

Trying to sort the coefficients of the polynomial $(z-a)(z-b)(z-c)…(z-n)$ into a vector

So I have a factored polynomial of the form $(z-a)(z-b)(z-c)\ldots(z-n)$ for $n$ an even positive integer. Thus the coefficient of $z^k$ for $0 \le k < n$ will be the sum of all distinct $n-k$ ...
14
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4answers
387 views

If $x$ and $y$ are rational numbers and $x^5+y^5=2x^2y^2,$ then $1-xy$ is a perfect square.

Prove that if $x, y$ are rational numbers and $$ x^5 +y^5 = 2x^2y^2$$ then $1-xy$ is a perfect square.
0
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1answer
147 views

Value $\Phi_n(1)$ of the cyclotomic polynomial at x=1 [duplicate]

Possible Duplicate: Value of cyclotomic polynomial evaluated at 1 I have to show $\Phi_n(1)=1$ for $n\neq p^k$ with $p$ is prime. (I already proved to easy part $\Phi_n(1)=p$ for $n=p^k$) ...
1
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1answer
97 views

Multiplying polynomials

Let $f(x)$ be degree $n$ polynomial, with $n+1$ nonzero monomial, i.e., all coefficients nonzero (for example if $n = 3$, then we could have $3x^3 + 2x^2 + x + 10$) Let $g(x)$ be any polynomial of ...
4
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2answers
283 views

Showing polynomials in $k[x_1, \ldots , x_n]$ are irreducible

It is often the case when I wish to show a particular polynomial in $k[x_1, \ldots ,x_n]$ is irreducible. Assuming that the polynomial is sufficiently friendly (i.e. one I would encounter as part of a ...
3
votes
1answer
109 views

Polynomial with a root that occurs n times, the root must be 0

I'm having a bit trouble with this excercise: The problem: Let there be a polynomial $f(x)=a_1x^{t_1} + a_2x^{t_2} + ... + a_nx^{t_n}$ Where $t_1, t_2, ..., t_n$ are not-negative integers. The ...
4
votes
3answers
1k views

How to tell if a quartic equation has a multiple root.

Is there any way to tell whether a quartic equation has double or triple root(s)? $$x^4 + a x^3 + b x^2 + c x + d = 0$$
1
vote
1answer
133 views

Is there a method for finding the common factor between two polynomials?

For two polynomials, there exist a method to find their monic GCD by a variation of Euclid's algoritm, is there any method exist for finding the exact nonmonic factor between two polynomials? ...
5
votes
3answers
141 views

Is this action of $\mathbb F[[x]]$ on $\bigoplus_{i=0}^{\infty}\mathbb F$ natural?

The title of my question has a field $\mathbb F$ in it, but to make sure I'm not losing anything, I would like to introduce my question in full generality. But still, I will be happy with an answer in ...
0
votes
1answer
70 views

Graphing polynomials

I'm trying to learn to factor polynomials, and I'm trying to graph $$30x^5 – 166x^4 – 542x^3 + 2838x^2 + 1520x – 800$$ from here on Purplemath. When I graph polynomials on my NSpire CS CAX or ...
11
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3answers
397 views

$f(f(x))$ has no fixed points if $f(x)$ has no fixed points

Assume that $f(x)=x$ has no real roots where $$f(x) = ax^2+bx+c$$ Prove that $f(f(x))=x$ has no real roots as well. What I've done is, calculating $f(f(x))$: ...
6
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0answers
119 views

On the order of $\mathbb{Z}[X]/(f,g)$ and the resultant $R(f,g)$.

I suspect that $\#\mathbb{Z}[X]/(f,g)=|R(f,g)|$ holds for any two non-constant polynomials $f,g\in\mathbb{Z}[X]$, where $R(f,g)$ is the resultant of $f$ and $g$. I am however unable to prove it. I'd ...
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1answer
86 views

Greater than zero?

I need to show that $$\sum_{i=k^*}^K\binom{K}{i}a^{i-1}(1-a)^{K-i-1}(i-aK)>0$$ given $K\geq k^*$, $0<a<1$ and $K$, $k^*\in\mathbb{Z^+/1}$. I did some computer simulation and saw that it ...
5
votes
1answer
175 views

Root of a special polynomial

Given a polynomial $P(x)=\sum_{n=0}^{d}a_nx^n\in\mathbb{R}[x]$ with all roots on the unit circle. Question: Is it true that all the roots of $Q(x)=\sum_{n=0}^{d}a_n{{x+d-n}\choose{d}}$ lie on a ...
2
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2answers
288 views

What is Hilbert polynomial of this projective variety?

Suppose you have a map $\varphi\colon\mathbb{C}^m\times\mathbb{C}^n\to\mathrm{Mat}_{m,n}(\mathbb{C})$ defined by sending $(\mathbf{u},\mathbf{v})\mapsto\mathbf{u}\cdot\mathbf{v}^T=(u_i,v_j)$. So ...
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vote
2answers
62 views

Optimization problem defined by polynomials only always leads to algebraic solutions?

Let $\Omega$ be a non-empty set in ${\mathbb R}^n$ defined by a set of polynomial inequalities with rational coefficients $P_i(x_1, \ldots ,x_n) \gt 0 (1 \leq i\leq m)$ and $Q_j(x_1, \ldots ,x_n) \geq ...
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0answers
82 views

GCD of a bivariate polynomial and its partial derivative..

I am stuck in the following question :- $f(x, y)$ is a bivariate polynomial with coefficients in $Z$. We have to show that $deg(GCD(f, f_y)) > 0$ iff $deg(GCD(f, f_x)) > 0$.(Here $f_x$ denotes ...
2
votes
1answer
99 views

Any endomorphism of $K[x]$ which is the identity on $K$ is $E_g$ for some $g$ in $K[x]$

Prove: Any endomorphism of $K[x]$ which is the identity on $K$ is $E_g$ for some $g$ in $K[x]$. Note: In my book it defines $E_g\colon K[x] \to K[x]$ by sending $x$ to $g$. This seems like it ...
3
votes
4answers
385 views

Factoring $ac$ to factor $ax^2+bx+c$

I was watching a first-year high-school-algebra student struggle with factoring quadratics last night. Given a quadratic $ax^2+bx+c$ (I'll give you the exact example in a moment), her method — ...
4
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1answer
330 views

How to incorporate erasures (known error locations) in computation Reed-Solomon error locator?

I'm implementing Reed-Solomon error correction for 2D barcode formats (part of the ZXing project). It already has a working implementation, which I managed to create, mostly years ago when I ...
5
votes
2answers
204 views

Proving a polynomial inequality over the real numbers. $x^{16}-x^{11}+x^6-x+1>0$

Prove that $x^{16}-x^{11}+x^6-x+1>0$ for $x\in R$. So I thought of something like this: $$x^{10}(x^6-x)+x^6-x>-1$$ $$(x^{10}+1)(x^6-x)>-1$$ But it seems to not be too much of help. While ...
2
votes
2answers
144 views

The roots of $t^5+1$

Just a quick question, how do we go about finding the roots of $t^5 +1$? I can see that since $t^5=-1$ that an obvious root is $\sqrt[5]{-1}$. I am assuming that since there is a $-1$ involved, some ...
1
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0answers
119 views

Extending Hermite polynomial interpolation

Working with the definition of Hermite polynomials $x_0,\ldots,x_n$ are distinct in $[a, b]$, $f''(x)$ is continuous on [a, b], then $$H_{2n+1}(x)=\sum_{j=0}^{n} [f(x_j)H_{n,j}(x)] +\sum_{j=0}^{n} ...
9
votes
3answers
338 views

Reducibility of $P(X^2)$

This question is inspired by a comment discussion in If $K=K^2$ then every automorphism of $\mbox{Aut}_K V$, where $\dim V< \infty$, is the square of some endomorphism.. Let $k$ be a field of ...
6
votes
1answer
199 views

Is $x^n+px+p^2$ irreducible in $\mathbb{Z}[x]$?

If $p\in\mathbb{N}$ is a prime, is $x^n+px+p^2$ irreducible in $\mathbb{Z}[x]$? I've proved that any non-unit factor in $\mathbb{Z}[x]$ must have degree at least 2. Eisenstein's criterion doesn't ...
1
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0answers
109 views

Unit group of quotient of noncommutative polynomial ring

In this recent post the original question led people to look for rigid, noncommutative rings. (Rigid means that the only endomorphisms are zero and the identity). Several (somewhat complicated) ...
1
vote
2answers
135 views

Rolle's theorem for showing that $(x-a)^k$ divides $p(x)$ . . .

Have the following I'm stuck on: Suppose $p(x)=p_0+p_1 x+p_2 x^2+\cdots+p_n x^n$ is a polynomial of degree $n \geq 1$. Show that if $(x-a)^k$ divides $p(x)$ for some $a\in\mathbb R$ and some ...