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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4answers
38 views

Find the remainder when $(x+1)^n$ is divided by $(x-1)^3$

Find the remainder when $(x+1)^n$ is divided by $(x-1)^3$ I know that \begin{equation*} (1 + x)^n = 1 + nx +\frac {n(n-1)}2!\cdot x^2 +\frac {n(n-1)(n-2)}3! \cdot x^3 +... \end{equation*} ...
4
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0answers
19 views

Remainders of quadratic trinomial

The problem is to determine, whether there exist a quadratic trinomial $f(x) = ax^2 + bx +c$ with integer coefficients (with $a$ not a multiple of 2014), such that the numbers $\ f(1), \ f(2),\, ...
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0answers
24 views

Factorisation of large polynomials and Galois theory

As I understand it, one of the consequences of Galois theory is that there is no way of expressing the solutions to a general polynomial of degree 5 or higher in terms of radicals. Would a theory that ...
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3answers
62 views

If a quadratic equation $ax^2+bx+c=0$ has more than two roots, then $a=b=c=0$ [on hold]

If a quadratic equation $ax^2+bx+c=0$ has more than two roots, then it is an identity i.e. it is true for all values of $x$ and $a=b=c=0$. What is a proof of this?
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1answer
53 views

what is the value of $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$?

if we have $a+b+c=1$ and $ab+bc+ac=\frac{1}{3}$ then what is the value of $$\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$$ and $$\frac{a}{b+1}+\frac{b}{c+1}+\frac{c}{a+1}$$. from the hypothesis we have ...
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0answers
21 views

Using Polynomial divison , the factor theorem and resolving into partial fractions simplify:

Thank you for any help given. I have been tryin to solve it for hours.
4
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2answers
44 views

Name for the following set of polynomials

I have the following set of polynomials defined by $$P_n(x) = \sum^n_{k = 0} \frac{n!}{k!} x^k, \quad x \geqslant 0.$$ The first few are \begin{align*} P_0 (x) &= 1\\ P_1 (x) &= 1+x\\ P_2 (x) ...
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3answers
61 views

Finding value of a quadratic

The polynomial \begin{equation*} p(x)= ax^2+bx+c \end{equation*} has $1+\sqrt{3}$ as one of it's roots and also $p(2)=-2$. Is there any way to know the value of $a$, $b$ and $c$? I tried but I can ...
1
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1answer
12 views

Is there a way to find a good lower bound on $\Vert p_n \Vert_\infty$ without finding the extrema?

Let $$p_n(x):=x^n+c_{n-1}x^{n-1}+ \cdots + c_0$$ be defined over some interval $[a,b]$. Is there a way to find a good lower bound on $\max_{x\in [a,b]} | p_n (x) |$ without actually finding the ...
2
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1answer
31 views

what is the decomposition of $(x^2+y^2+z^2)(x+y+z)(x+y-z)(-x+y+z)(x-y+z)-8x^2y^2z^2$?

I want to have a decomposition of this : $$(x^2+y^2+z^2)(x+y+z)(x+y-z)(-x+y+z)(x-y+z)-8x^2y^2z^2$$ I have tried all possible calculation which came to my mind,I will describe one of it which is ...
3
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1answer
38 views

Polynomials and Commutativity

Let $f(x)=2013x+1$. Suppose $g(x), h(x)$ are polynomials with real coefficients such that $f(g(x))=g(f(x))$ and $f(h(x))=h(f(x))$. Prove that $g(h(x))=h(g(x))$. I tried to look at the coefficients of ...
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2answers
50 views

Find the lowest degree of the polynom $P$?

I have to determine the lowest degree of $P$ given by the following system : $\left\{ \begin{array}{l} P \equiv 2X \ \mod[X^2 -2X +1] \\ P \equiv 3X \ \mod[X^2 -4X+4] \end{array} \right.$ First, ...
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2answers
36 views

How to go about this proof for non zero polynomials.

How do I go about proving this? Let $\mathbb{F}$ be a field and $X$ an indeterminate, and consider the polynomial ring $\mathbb{F}[X]$. Let $f(X), g(X) \in \mathbb{F}[X]$ with $f(X), g(X) \neq ...
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0answers
17 views

When is ${(1-t^2)^{-N/2}}{\det(f_t(A))}$ expressible as polynomial?

Given a matrix valued function $f_t:\mathbb R^{N\times N} \mapsto \mathbb R^{N\times N}$. For $f_t(A)=B$ both $A$ and $B$ are symmetric. Which properties can be assigned to $f_t$, when the following ...
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3answers
49 views

A question about polynomials [duplicate]

How can we prove that the following expression is a polynomial? $$ \frac{1-x^{2^{n-1}}}{1-x} $$ I've asked this question just for learning the ways different from using ...
2
votes
2answers
45 views

Real points of order 3 on an elliptic curve.

This comes from Silverman's Rational Points on Elliptic Curves: Consider the elliptic curve (non singular) $y^2=x^3+ax^2+bx+c=f(x)$ after some computations we can see that points of order 3 in this ...
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0answers
10 views

LFSR - feedback polynomial

I want to describe the recursion S$_{t}$=S$_{t-2}$+S$_{t-3}$ with help of a Trace function in $\mathbb{F}_{2}$. I found the feedback polynomial f(x) = $x^3+x+1$ But how to continue ? How can I find ...
2
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4answers
54 views

Division by zero factoring problem

Given the fact that the function: $$f(x) = \dfrac{2x-4}{x-2} = \dfrac{2(x-2)}{x-2} = f(x) = 2$$ Shouldn't $f(x)$ be equal to $2$ in every case, even when $x=2$?
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2answers
47 views

Number of roots of a polynomial of non-integer degree

How many roots does a polynomial-like function of degree $n$ have if $n$ is a rational or an irrational number?
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1answer
31 views

Square root of an even polynomial is holomorphic

Given an even degree polynomial $p(x)$, all of whose roots satisfy $|z| < R$. Explain why there is a holomorphic (i.e. analytic) function $h(z)$ defined on the region $R < |z| < ∞$ which ...
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1answer
24 views

Polynomial irreducibiliy with substitution (need evaluation of logic)

One thing I have seen several times when trying to show that a polynomial $p(x)$ is irreducible over a field $F$ is that instead of showing that $p(x)$ is irreducible, I am supposed to show that $p(ax ...
2
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1answer
33 views

If some non-primitive polynomial in $\mathbb Z[x]$ is irreducible over $\mathbb Q$, does this imply it is irreducible over $\mathbb Z$?

I know from a lemma in Herstein that if a primitive polynomial in $R[x]$ is irreducible in its field of quotients $F[x]$, then it is irreducible in $R[x]$. But, if some non-primitive polynomial in ...
2
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1answer
33 views

Division algorithm for polynomials

When we do the division algorithm for polynomials, how do we figure out $ca^{-1}$; i.e., for the problem where $f(x)= 3x^2+2$ and $g(x)= 4x^4 + 2x^3 + 6x^2 + 4x + 2$ in $\mathbb{Z}_7[x]$. Here, $a= ...
10
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1answer
184 views

Solving a special Quartic Equation.

Solve for $x$ $$(x^2-4)(x^2-2x)=2$$ I have tried the Rational Root Theorem and found that there are no rational roots. Further, the polynomial $p(x)=(x^2-4)(x^2-2x)-2$ is irreducible since ...
0
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1answer
10 views

cubic approximation with four points(approximating sine function with polynomials)

I was reading the following article https://mixedmath.wordpress.com/2013/11/17/an-intuitive-overview-of-taylor-series/ regarding Taylor Series.When I got to the part 1.3. Cubic approximation I got ...
2
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0answers
23 views

Possible values of the GCD of two polynomials

Let $p(x)$ be a polynomial in $\mathbb Q[X]$. Find the possible values of $d=gcd(p(x),p(x)+x-1)$. I have: $gcd(p(x),p(x)+x-1)=gcd(p(x),x-1)$ Is the answer to the question: $d(x) \in ...
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2answers
42 views

Why is the remainder of any polynomial divided by a 1st degree polynomial, a constant

Here is a "Math is fun" quote: "When we divide by a polynomial of degree $1$ (such as "$x-3$") the remainder will have degree $0$ (in other words a constant, like "$4$")." I'm hoping someone could ...
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2answers
26 views

Quotient rings, polynomials are reducibility

I am trying to follow this solution. I am struggling to understand why 'If g is a member of R, then g divides the content of f'. Why is this true?
1
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1answer
27 views

Show that a real polynomial of degree more than or equal to $3$ is reducible

Let $f \in \mathbb R [x]$ and suppose that $\deg(f) \geq 3$. Then $f$ is reducible. Proof: By the Fundamental theorem of algebra there are $\lambda _j \in \mathbb C$ such that $$f(x) = (x-\lambda_1) ...
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0answers
20 views

Determine the order of $\exp Q(z)$ when $Q$ is a polynomial of degree $q$.

I am looking to determine the order of $f(z) = \exp Q(z)$ when $Q$ is a polynomial of degree $q$. I think the order is $q$, but I am struggling to prove it. The definition of order is: An entire ...
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0answers
32 views

What are the solutions to this equation? (hyperboloid)

Equation: $$(x - y - z ) A - (x^2 - y^2 - z^2)=0$$ I am trying to find all the possible solutions for the equation above. $A$ is a real strictly positive constant , $A>0$. $x,y,z$ are non ...
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1answer
53 views

Roots of field of polynomial [on hold]

Let $F$ be a field. And let $\alpha ,\beta \in \mathbb F$ are roots of $a + bx +cx^2 \mathbb \in F[x]$ with $c\neq0$, then show that $\alpha +\beta =-bc^{-1} $ and $\alpha \beta = ac^{-1}$. I'll be ...
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0answers
30 views

Finding sum of coefficients of even powers of a polynomial [on hold]

Just how am I supposed to denote the sum of coefficients even powers of a polynomial f(x)
6
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1answer
44 views

Gauss's lemma: More than a stepping stone on the way to proving $R[x]$ is a UFD when $R$ is?

I'm reviewing my abstract algebra a bit. Currently looking at UFDs. In this context, Gauss's lemma (or part of it, at least) says that the product of two primitive polynomials over a UFD is primitive. ...
2
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0answers
31 views

Is my understanding of this corollary correct?

The following is a theorem/corollary pair in an introductory abstract algebra course. Theorem: $f(x)\equiv g(x) $ mod $p(x)$ if and only if $[f(x)]=[g(x)]$, where $[h(x)]=h(x)$ mod $p(x)$. ...
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2answers
63 views

polynomials root finding [on hold]

Is every root of a polynomial of positive integer degree n, and with a rational coefficients is considered algebraic number? and how one can find some roots to this polynomial ...
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0answers
77 views

Centripetal Catmull–Rom spline

What is "t" in this short and simple example below? There are 4 points Pn[xn,yn] in 2D space: A[1,6] B[3,1] ...
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0answers
8 views

Representation of polynomial order in CFD codes

I currently working on a CFD code over a cubic grid. Now, the number of elements used in the simulation is decomposed among the number of processors. Each of those processors (a section of the cube) ...
2
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1answer
99 views

Irreducible Polynomials over Finite Fields [on hold]

How would I show that $p(x)=x^5+x^2+1$ is an irreducible polynomial over $\Bbb Z_2=\{0,1\}$.
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0answers
24 views

Explain why $I$ is a function from $P$ to $P$ and determine whether it is one-to-one and onto.

The question and the solution are:( uploaded a photo so it is easier to see the formulas) So I am confused about the formula of p(x). P is the set of polynomial of x. OK, but why it makes p(x) = ...
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1answer
32 views

Regularity of a quotient ring of the polynomial ring in three indeterminates

Let $I=(f)$ be a prime ideal in $R=\mathbb{C}[x,y,z]$, so $f$ is an irreducible polynomial, and further assume that $f$ is of the following form: $f=z^n+c_{n-1}z^{n-1}+\ldots+c_1z+c_0$, where ...
2
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1answer
37 views

Constant term of noncommutative $(X+Y+(XY)^{-1})^n$

As the title reads I am trying to find a formula for the constant term of the above noncommutative polynomal expression, $$[1](X+Y+(XY)^{-1})^{3n}\quad \bigg(\in \mathbb{C}\langle X^{\pm 1},Y^{\pm ...
3
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4answers
45 views

Simplifying quartic complex function in terms of $\cos nx$

$$z= \cos(x)+i\sin(x)\\ 3z^4 -z^3+2z^2-z+3$$ How would you simplify this in terms of $\cos(nx)$?
12
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1answer
172 views

Polynomials $f$ with integer coefficients such that $f(x) \geq 0$ on $[-2,2]$ and $f(x) \leq \frac{1}{1+x}$ on $(-1,2]$

Find polynomials with integer coefficients $f\in\mathbb{Z}[x]$ such that $f(x)\ge 0$ on $x\in[-2,2]$ and $\frac{1}{1+x}\ge f(x)$ on $x\in(-1,2]$. I guess only such polynomial is just $0$, but it ...
0
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1answer
25 views

minimal value polynomials with integer coefficients

Let $D$ be the set of polynomials of integer coefficients $f\in\mathbb{Z}[x]$ such that $f(x)\ge 0$ at $x\in[-2,2]$, where the zero polynomial $f=0$ is excluded. Can I find a finite "minimal" set ...
0
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0answers
19 views

Barbeau's Polynomials: Quadratic Polynomials, 1.2.2

I've verified $(a)$ by expanding the $RHS$. I've partially verified $(b)$ doing the following: $$\begin{eqnarray*} ...
0
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3answers
64 views

Having the roots of a polynomial, is it possible to go back and find a polynomial that have exactly these roots?

This might be very silly. But I've been wondering if it's possible to assume $n$ numbers as roots of $p(x)$ and find a polynomial that have these roots. I've made a table with some polynomials ...
1
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1answer
37 views

Interpretation of summations in regards to combinatorics

I've been studying for a final in combinatorics and ran into 3 different summations that have me stumped. 1) interpret the equation in terms of counting words. (Hint: $e^a$$e^b$$e^c$) $$e^{3x} = ...
1
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3answers
52 views

How to find the remainder of polynomial division?

Im trying to solve this problem but I do not understand what the question is asking: Let $n\ge 2$ be an integer and $ p_n(x) $ be the polynomial: $$ p_n(x) = (x-1)+(x-2)+\cdots+(x-n) $$ What is the ...
1
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4answers
147 views

Show a polynomial is irreducible mod 29

Is there an easy way to see that the polynomial $x^2 + 3x + 10$ is irreducible modulo 29 without having to go through each element 0,1,..,28 and check for roots?