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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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 ...
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1answer
35 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 ...
5
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1answer
920 views

Factorization of $x^7-1$ into irreducible factors over $GF(4)$

I need to find cyclotomic cosets depending on $n=7$ and $q=4$ and find the factorization of $x^7-1$ into irreducible factors over $GF(4)$. Thanks for any advice.
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1answer
36 views

Methods for factorising polynomials into irreducibles over finite fields

I was given a problem recently, part of whose solution was to factorise $x^{15}+1$ in $\mathbb F_2[x]$. It turns out that the factorisation is $$(x+1)(x^2+x+1)(x^4+x^3+x^2+x+1)(x^4+x+1)(x^4+x^3+1),$$ ...
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1answer
192 views

How many quadratic polynomials exist given the two zeroes? ($1$ or $\infty$)

I was reading some book which had this question: Q. The number of [quadratic] polynomials having zeros $-2$ and $5$ is: (A) 1 (B) 2 (C) 3 (D) More than three? Sol. (A) 1. But ...
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= ...
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1answer
186 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 ...
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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 ...
12
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1answer
320 views

Krylov-like method for solving systems of polynomials?

To iteratively solve large linear systems, many current state-of-the-art methods work by finding approximate solutions in successively larger (Krylov) subspaces. Are there similar iterative methods ...
6
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1answer
45 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. ...
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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 ...
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1answer
54 views

Roots of field of polynomial [closed]

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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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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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
321 views

How does the cross multiplication of Quadratic Equation work?

How does the cross multiplication of Quadratic Equation works? If: $$f_1\left(x\right)=a_1x^2+b_1x+c_1=0$$ and: $$f_2\left(x\right)=a_2x^2+b_2x+c_2=0$$ have a common root, let's say, $\alpha$, then ...
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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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?
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0answers
22 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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1answer
19 views

Series expansions of inverse polynomials

Suppose one is given a strictly monotonous polynomial, $$f(x) = \sum_{n=0}^N a_n x^n$$ So that for a given $y$ there exists a single real $x=f^{-1}(y)$. It would be nice* to be able to calculate the ...
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2answers
120 views

Degree of polynomial :$ [x + (x^3 – 1)^{1/2}]^5 + [x – (x^3 – 1)^{1/2}]^5 $

What is the degree of the polynomial of the following expression ? $$ [x + (x^3 – 1)^{1/2}]^5 + [x – (x^3 – 1)^{1/2}]^5 $$ If I am not very wrong the highest power of x is $\frac {15}{2} $ ?! So ...
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0answers
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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) ...
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0answers
32 views

Finding sum of coefficients of even powers of a polynomial [closed]

Just how am I supposed to denote the sum of coefficients even powers of a polynomial f(x)
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0answers
32 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 $p(x)$. ...
12
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1answer
173 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 ...
132
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18answers
14k views

Why can ALL quadratic equations be solved by the quadratic formula?

In algebra, all quadratic problems can be solved by using the quadratic formula. I read a couple of books, and they told me only HOW and WHEN to use this formula, but they don't tell me WHY I can use ...
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0answers
25 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) = ...
8
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1answer
82 views

If all convex combinations of $p(x)$ and $q(x)$ have real roots, then $p,q$ have a common interlacing poly

I heard this result in a talk the other day: Suppose $p$ and $q$ are polynomials. Suppose $p$ is a polynomial of degree $n$ and $q$ a polynomial of degree $n-1$. Call $q$ and interlacer of $p$ if the ...
2
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1answer
103 views

Irreducible Polynomials over Finite Fields [closed]

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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9answers
358 views

Prove that $f=x^4-4x^2+16\in\mathbb{Q}$ is irreducible

Prove that $f=x^4-4x^2+16\in\mathbb{Q}[x]$ is irreducible. I am trying to prove it with Eisenstein's criterion but without success: for p=2, it divides -4 and the constant coefficient 16, don't ...
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4answers
114 views

Show that the ideal of $k[X_1, X_2, X_3]$ generated by $X_1^3-X_3$ and $X_2^2-X_3$ is a prime ideal.

Show that the ideal $I$ of $k[X_1, X_2, X_3]$ generated by $X_1^3-X_3$ and $X_2^2-X_3$ is a prime ideal, where $k$ is a field. I tried to prove it by contradiction. Suppose $f$ and $g$ are not of ...
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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 ...
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} = ...
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2answers
512 views

Can you help me reverse the Minimum Curvature Method?

The minimum curvature method is used in oil drilling to calculate positional data from directional data. A survey is a reading at a certain depth down the borehole that contains measured depth, ...
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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 $\cos(nx)$?
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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 ...
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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*} ...
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3answers
66 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 ...
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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 ...
4
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1answer
43 views

Trigonometric root of a polynomial

If $4\cos^2 \left(\dfrac{k\pi}{j}\right)$ is the greatest root of the equation $$x^3-7x^2+14x-7=0$$ where $\gcd(k,j)=1$ Evaluate $k+j$ I tried factorizing the equation but it wasn't ...
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1answer
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Isomorphism of polynomial rings [closed]

How I prove, $\mathbb{R}/(x^2)$ is isomorphic to Dual number Ring. Dual Number ring The dual number ring is define by $D={a+bϵ:a,b∈R,ϵ^2 =0}$
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3answers
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What are the values of $p$ so that equation $x^3+(p-2)x^2+(5-2p)x-10=0$ has exactly $2$ real roots…

I found this question in a maths-group in Facebook- What are the values of $p$ so that equation $x^3+(p-2)x^2+(5-2p)x-10=0$ has exactly $2$ real roots........ I think we do not count repeated roots ...
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2answers
424 views

Application of Taylor's Theorem in Number Theory

I'm working through Alan Baker's book A Concise Introduction to the Theory of Numbers, and there's an assertion in there that confuses me. Here's the quote: It is easily seen that no polynomial ...
3
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3answers
110 views

Prove $f=1+x+x^2+x^3+\cdots+x^n$ has no multiple roots.

Prove $f=1+x+x^2+x^3+\cdots+x^n$ has no multiple roots. My attempt: Consider the polynomial $g=(x-1)(1+x+x^2+x^3+\cdots+x^n)$ As $f\mid g, g$ all the roots of $f$ are roots of $g$. This means I ...
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1answer
20 views

Find the coordinate matrix of a polynomial with respect to a non-standard basis

I'm stuck on this question here: Find the coordinate matrix of $2-4x-3x^2$ with respect to $B = {2, x^2-1, 1-2x-x^2}$ I did the following: $a(2) + b(x^2 - 1) + c(1-2x-x^2) = 2-4x-3x^2$ But now I'm ...
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1answer
30 views

Sylvester's dialectic method

Nevermind. I have got it I am willing to study Sylvester's Dialectic Method regarding polynomial. Today in one book I have come to know about it but could not find it. In MSE I tried my best to find ...
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2answers
149 views

A variant of the Schwartz–Zippel lemma

Let $f \in \mathbb{F}[x_1,\ldots,x_n]$ be a nonzero polynomial. Let $d_1$ be the maximum exponent of $x_1$ in $f$ and let $f_1$ be the coefficient of $x_1^{d_1}$ in $f.$ Let $d_2$ be the maximal ...
0
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3answers
38 views

Factor this polynomial into linear factors with coefficients in $F = \mathbb{Q}(2^{1/3}, i\sqrt{3})$

The polynomial is this: $x^3 -2$ Okay, so first I can create my field extension. I can easily extend the field to $2^{1/3}$. And I know the elements of the extension of $\mathbb{Q}(2^{1/3})$ can be ...
3
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1answer
54 views

Working in $\mathbb Q[x]$. Two polynomials are coprime if their gcd is a constant?

When are two polynomials coprime? Is it when their gcd is a constant? If we divide $x^3-7x-5$ by $x-4$, we get: $$x^3-7x-5=(x-4)(x^2+4x+9)+31$$ So, is $31$ their gcd, but since $31$ is not monic ...
1
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1answer
26 views

a question about field theory and polynomials

Hello all I was given this question in my field theory class on which I would certainly appreciate the help: I am given a field F of characteristic p ($ ch(F) > 0 $) and this polynomial $ f(x) = ...