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 function with 2 points

Can someone please help me solve this problem ? ...
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3answers
71 views

Roots of $x^2 +2x +2$ Over $\mathbb{C}$

Find the roots of $x^2 +2x +2$ over $\mathbb{C}$ I need to prove somehow that the roots will be $(1 + i) , (1 - i)$ Any ideas how can I find those roots in a simple way?
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Using Remainder or Factor Theorems to Find Coefficient

I'm studying the remainder and factor theorems and a question asks: -4 is a root of $x^4 + ax^3 - 19x^2 - 46x + 120$ What is the value of a? Since -4 is a root then I can deduce that x+4 is a ...
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3answers
76 views

Prove there exists a unique $n$-th degree polynomial that passes through $n+1$ points in the plane

I know given two points in the plane $(x_1,y_1)$ and $(x_2,y_2)$ there exists a unique 1st degree (linear) polynomial that passes through those points. We all learned in Algebra how to find the slope ...
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2answers
124 views

help interpreting an abstract algebra test question

This is a take-home test problem, and I don't want help solving it, just understanding what it's asking. I've asked my prof a couple times, but she's either unwilling or unable to give me a straight ...
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1answer
55 views

How many natural value of n such that $n^5+2n^4+n-1$ is prime number?

From above polynomial, I can only get one value to make it prime. The value, I guess, is only one. For $n=1$, we got: $$(n^5+2n^4+n-1)= 1+2+1-1= 3 \quad\text{(prime)}$$ But, I cannot find the ...
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1answer
34 views

Lower bound for $(x + y)^k $?

I'm wondering, is there a lower bound for $(x + y)^k $? For example, if $x,y,k > 0$, can we say that $(x + y)^k \geq x^k + y^k$? If anyone has a source/reference for this, that would be great.
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1answer
38 views

Show that $\frac p{a^n}-\frac q{b^n}-1=0$.

If $x^n-py^n-qz^n$ is exactly divisible by $x^2-(ay+bz)x+abyz$ , then prove that $\frac p{a^n}-\frac q{b^n}-1=0$
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1answer
41 views

Irreducible factor decomposition

This is a past exam question. Decompose each of the following elements as a product of irreducible: (a) $X^4+2 \in \mathbb{Z}_5[X]$ (b) $X^5+X \in \mathbb{Z}_2[X]$ (c) $X^5+4X^4-3X^3+X^2+7X+11 \in ...
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0answers
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Polynomial addition using linked lists (theory only)

Someone on Stackoverflow suggested posting here. I have an Algorithms test coming up for my Software Engineering course, but the book and lecture notes don't explain the theory behind summing ...
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0answers
20 views

Field extension of fraction field of polynomial ring modulo an ideal.

My apologies for the relatively long question, but I am trying to understand a step in a proof, which needs some preliminary explanation. Let $K$ be a field and $I$ a prime ideal of ...
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1answer
29 views

Irreducible in $\mathbb{Z}[X], \mathbb{Q}[X], \mathbb{R}[X]$ and $\mathbb{C}[X]$

Is the polynomial $2X^3-10X^2+50X+10$ rrreducible in $\mathbb{Z}[X], \mathbb{Q}[X], \mathbb{R}[X]$ and $\mathbb{C}[X]$. What I have done so far is, since this is a cubic function, therefore it will ...
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3answers
40 views

Explanation on characterstic polynomial

$A_2 = \begin{pmatrix} 1 & 1 \\ a & 1 \end{pmatrix} $ So the characteristic polynomial of $A_2$ is $P_a(t) = (t-1)^2 - a $ Then, $ P_a(t) = t^2 -2t +1 -a$ ...
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1answer
108 views

There cannot exist a rational function $f: \mathbb{R} \to \mathbb{R}$ injective, not surjective

I was looking for a rational function $f: \mathbb{R} \to \mathbb{R}$ that looks like $\arctan$, in that it is injective not surjective well-defined on all $x\in \mathbb{R}$ (no vertical ...
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1answer
36 views

GCD of polynomials over $\mathbb{Z}_3$

$f$ and $g$ are polynomials over field $\space \mathbb{Z}_3$. $f=X^4+X^3+X+2, \space g=X^4+2X^3+2X+2$. And I been asked to find the GCD of them. What I have done is using Euclidean algorithm. After ...
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0answers
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Find canonical form of bi-linear form on polynomials

$V$ is vector space of polynomials in degree less or equal than $2$, we define the bi-linear form: $f(p,q) = p'(-1)q(2$) where $p$ and q are polynomials from $V$. I need to find the canonical form ...
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1k views

Is this a polynomial?

$$x^4 + x^3 + x + 1$$ Notice how I skipped $x^2$. Do "polynomials" need to have a sequence of exponents that start from $1$ and go up by $1$ and only $1$ each time? Thanks
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1answer
52 views

Prove that $x^2+1$ is reducible in $\mathbb{Z}_p[x]$.

Prove that $x^2+1$ is reducible in $\mathbb{Z}_p[x]$ if and only if there exists integers $a$ and $b$ such that $a+b=p$ and $ab \equiv 1 \pmod{p}$. (Here $\mathbb{Z}_p$ means the integers modulo a ...
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1answer
50 views

Derivation and application of Newton's identity

How is the following identity derived? $$\sum_{\ell =0}^{n-1}(-1)^\ell e_\ell s_{n-\ell}+(-1)^nne_n=0$$ Is there an example demonstrating the context in which this might be applied?
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2answers
404 views

How can I find the roots of a quartic equation, knowing one of its roots?

I need to decompose (in $\Bbb{C}[x]$) the function $$ f(x) = x^4 + 4x^3 - 4x^2 + 24x + 15 $$ in its simplest form, knowing that $1 - 2i$ is one of its roots. Any ideas?
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1answer
23 views

Showing $f\in\mathbb{F}_{p^d}[X]:f'=0\Rightarrow\exists g\in\mathbb{F}_{p^d}[X]:f=g^p$

Let $\mathbb{F}_{p^d}$ denote the final field with $p^d$ elements and $\mathbb{F}_{p^d}[X]$ denote the polynomial ring in $X$ over $\mathbb{F}_{p^d}$. How can we show ...
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2answers
36 views

Possible value of function

Let $f(x) = x^3 + ax^2 + bx + c$. Here, $a,b,c$ are integers. If $f(1)=f(6)=0$ then find a possible value of $f(4)$. I do not know how to solve these type of problems. So can anyone please provide ...
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2answers
30 views

not complete polynomial and roots

It might be very simple but I need a formal proof for accepting or rejecting the idea below. Let g be a polynomial of the order of n given below $$ g(L)=1-\theta_1 L-\theta_2 L^2- \ldots -\theta_n ...
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1answer
21 views

Cardinality of a set with a recurrence relation.

Let $A = \left\{ f\in \mathbb{N}\rightarrow \mathbb{C} \mid \forall n\in \mathbb{N}. f(n+3) + 3f(n+1) = f(n+2)+f(n) \right\}$ What is $\left|A\right|$? Well, I tried to treat $f$ as a recurrence ...
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2answers
43 views

Help understanding the characteristic polynomial

Let $$A = \begin{bmatrix} 1 &2 &1 \\ 2 & 2 &3 \\ 1 & 1 &1 \end{bmatrix}$$ I'm calculating the characteristic polynomial by the following: $$P(x) = -x^3 + Tr(A)x^2 + ...
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1answer
45 views

Degree of Equations [closed]

A) Which variables in the formula $V = \pi r^2 h$ would you need to set as a constant in order to generate: a linear equation? a quadratic equation? B) How should $r$ and $h$ be ...
2
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1answer
89 views

Is the ring $ R = \{ f \in \mathbb{C}[x,y] \mid {\nabla f}(0,0) = (0,0) \} $ Noetherian?

Question: Is the ring $ R = \{ f \in \mathbb{C}[x,y] \mid {\nabla f}(0,0) = (0,0) \} $ Noetherian? I guess it isn’t Noetherian as I suspect that $$ (x y + y^{2}), \quad (x y + y^{2},x^{2} y + ...
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36 views

Finding a quadratic equation from roots (Vieta's formula)

Given $x_1 = 1-\sqrt 3$ and $x_2=1+\sqrt 3$, What is the quadratic equation? By Vieta's formula: $-\frac{b}{a} = 1-\sqrt 3 + 1+\sqrt 3 = 2$. Hence, $-b = 2a$ $-\frac{c}{a} = (1-\sqrt 3)(1+\sqrt 3) ...
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3answers
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Find the intersections of the functions

I have $f(x)=-x^2+4$ a parabola and $g(x)=\sqrt{(4-x^2})$ a semi circle with a raduis of $2$ if I say $g(x)=f(x)$ and solve for $x$. I should find the points at which $x$ intercepts ...
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1answer
56 views

What is a rational way of factoring polynomials?

Consider the following polynomial $$P_x:=x^4+1$$ I want to represent it as a product of two polynomials with real coefficients of grade $2$. I have done this using, call it, a brute force method. ...
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24 views

Polynomial Divison of unknowns

How to complete this $$P(X)=\frac{2x^4-7x^3+5x^2+ax+b }{ 2x^2+x-1}$$ so that the division is without a remainder? When it is divided it gives two equation.
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Coefficient of polynomials

Could someone explain to me why $$ [x^{24}](1-2x^6)^{-31} = 2^4 \binom{4 + 31 - 1}{31 - 1} \, ? $$ Reads: The coefficient of $x^{24}$ in $(1-2x^6)^{-31} =$ ...
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6answers
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Polynomial factors involving inequalities

How to factorise the polynomial $p(x) = x^4-2x^3 + 2x - 1$. Hence, solve the inequality $p(x) \gt 0$ ?
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1answer
79 views

Is there a simple procedure to produce algebraic numbers of modulus one that are not roots of unity?

Let $z=e^{i\theta}$ be a complex number of modulus $1$. Trivially if $z$ is a root of unity then $z$ is also an algebraic number, but the converse is known to be false : $z$ can be algebraic without ...
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1answer
57 views

Solving polynomial equations over finite fields

I have looked (a bit) at questions like finding the number of roots of $x^n =1$ over a finite field. Now I would like to understand how to solve polynomial equations over finite fields. From what I ...
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1answer
127 views

Will this sequence of polynomials converge to a Hermite polynomial pointwise?

While trying to solve this question my testing lead to an observation that I found interesting in its own right. Consider the linear transformation $L:P\to P$ from the space of polynomial functions ...
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2answers
65 views

Calculating the characteristic polynomial

I'm stuck with this problem, so I've got the following matrix: $$A = \begin{bmatrix} 4& 6 & 10\\ 3& 10 & 13\\ -2&-6 &-8 \end{bmatrix}$$ Which gives me the following ...
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1answer
31 views

Do these rules of factorization for polynomials in multiple variables [given evaluation data] work?

Let $K$ be a field of characteristic $0$. I want to know if either of the following rules hold in $K[X, Y]$. Let $p(X, Y) \in K[X, Y]$ be such that for all $a \in K$, $(X - a)$ divides $p(X, ...
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2answers
71 views

$f(x) = x^3 - x$ then $f(n)$ is multiple of 3

If $f(x) = x^3 - x$ then $f(n)$ is multiple of 3 for all integer $n$. First i tried $$f(n) = n^3-n=n(n+1)(n-1)\qquad\forall n\ .$$ When $x$ is an integer then at least one factor on the right is ...
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2answers
354 views

Does this polynomial have all its roots both distinct and real?

Recently, I wondered about the following problem: let $n\geq 5$ and let $$ P_n(x)=(x-1)(x-2)\ldots (x-n)-1 $$ Is it true that $P_n(x)$ has $n$ distinct real roots for any $n\geq 5$ ? I checked it ...
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2answers
71 views

Root of the polynomial $x(x-1)(x-2)\cdots(x-K)=C$

Is there an analytic way to obtain the highest root of the polynomial $x(x-1)(x-2)\cdots(x-K)=C$ in terms of $K$ and $C$? The integer $K \ll x$ and the constant $C$ are known. The other way to ask ...
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0answers
17 views

Finding matrix index from triangular array offset

I have a mapping from a lower triangular matrix, A, to a vector,v: A(i,j) -> v( $\lfloor i(i+1)/2 \rfloor + j$ ) $i,j\in[0,N]$, $j\leq i$, $N\in\cal{N}$, $N\geq 0$ (so, my first row is row 0, and ...
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0answers
27 views

Straightedge-Compass Construction vs Origami

I've been looking into these two types of geometric construction and I was wondering- why is origami capable of solving up to cubic equations, when straightedge-compass construction is only capable of ...
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1answer
20 views

Inverse functions of certain conformal maps

Suppose I have a family of conformal maps given by $\omega=z+az^n$ where $|a|\leq1/n$ which maps the unit disk onto the so called epitrochoid. I am wondering about the inverse conformal map which ...
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Polynomial Roots of Bivariates

I've got a few polynomials that I am trying to get some results for (shown below). They come from the characteristic equation of a matrix. I have two variables in the polynomials, $\eta$ (which is ...
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1answer
54 views

Understanding 2012 AMC 12B #23

Monic quadratic polynomial $P(x)$ and $Q(x)$ have the property that $P(Q(x))$ has zeros at $x=-23$, $-21$, $-17$, and $-15$, and $Q(P(x))$ has zeros at $x=-59$,$-57$,$-51$ and $-49$. What is ...
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1answer
37 views

Question about nonisomorphic polynomial rings.

Let $n,k > 1$ be positive integers. Define the reduced polynomial rings : $g^k_n = \Bbb R[X_n]/(G^k_n(X_n))$ where $G^k_n$ is a real polynomial of degree $n$ (that keeps the ring reduced). (k is ...
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0answers
42 views

Gosper summable

I'd like to know why the following is NOT gosper summable: $$\sum_{k\in \Bbb{Z}} \frac{p(k)}{\prod_{j=0}^{m-1}(k+a+j)}$$ where $m>0, m\in\Bbb{Z}$ and $p(k)$ is a polynomial of degree $k=m-1$.
0
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1answer
78 views

$­\prod_{k=1}^{n}(x_k-a)(x_k-b)\leqslant\sum_{q=1}^{n}x_q^2\prod_{p=1,p\neq q}^{n}(x_p-a)(x_p-b).$?

Is there a name for this formula? For $f_k,w_k\geqslant0$. $$­\prod_{k=1}^{n}f_k\leqslant\sum_{q=1}^{n}w_q\prod_{p=1,p\neq q}^{n}f_p.$$ I believe that there is $w_k$ that make the formula true. Am I ...
5
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0answers
120 views

A question in my mind

Suppose $\displaystyle P\in \mathbb{R}[x]$ such that : $\displaystyle P(x)=2^n$ has at least one rational root for each $n\in \mathbb{N}$. Does it follow $P$ is linear? If it does or doesn't give ...