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 of degree 5 divisible by other polynomials.

I need some help with a problem. Find a polynomial $f(x)$ of degree $5$ such that both of these properties hold: $f(x)-1$ is divisible by $(x-1)^3$. $f(x)$ is divisible by $x^3$. I can't seem to ...
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185 views

Polynomial f(x) degree problem.

Suppose the polynomial f(x) is of degree 3 and satisfies f(3)=2, f(4)=4, f(5)=-3, and f(6)=8. Determine the value of f(0). How would I solve this problem? It seems quite complicated...
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How to solve the inequality: $\prod_{k=1}^N\left(x^k-k^2\right)\gt0$

Given the inequality: $$\displaystyle\prod_{k=1}^N\left(x^k-k^2\right)\gt0$$ how can I solve it? I suppose there is a difference if $N=2n$ or $N=2n+1$ with $n\in\mathbb{N}$, but I'm unable to find a ...
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2answers
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A question about algebraically closed fields

A field $\mathbb{K}$ is said to be algebraically closed in practice if every polynomial over $\mathbb{K}$ of positive degree less than or equal to $10^{10}$ has zero belonging $\mathbb{K}$. The ...
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prove $\sum\limits_{cyc} \frac {a^3} {b+c+d} \geq \frac {1} {3}$

Show that if $a,b,c,d \geq 0$ and $ab+bc+cd+da=1$ :$$\sum\limits_{cyc} \frac {a^3} {b+c+d} \geq \frac {1} {3}$$ yet again it should be solved with Cauchy inequality. thing i have done so far: ...
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How to prove $\sum_{i=1}^k(\frac{1}{\alpha_i}\prod_{j\neq i}^k\frac{\alpha_j}{\alpha_j-\alpha_i})=\sum_{i=1}^k\frac{1}{\alpha_i}$?

How to prove $\sum_{i=1}^k(\frac{1}{\alpha_i}\prod_{j\neq i}^k\frac{\alpha_j}{\alpha_j-\alpha_i})=\sum_{i=1}^k\frac{1}{\alpha_i}$? Where $\alpha_1, \alpha_2,\ldots, \alpha_k$ are $k$ distinct ...
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Polynomial Division - “Define the largest natural number…” [on hold]

Would someone mind helping me with this question? The more detailed possible so I can have 100% of understanding. Thanks. Question: Define the largest natural number m such that the polynomial ...
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1answer
27 views

Sum of nth powers and generalized polynomial sum

So this is a 2-part question (both parts I believe are closely related): How exactly does on express the sum $$\sum_{i=0}^{k}{i^n} = Q(n,k)$$ in a closed form For arbitrary positive integers ...
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Please help me with this problem on binomial expansion [on hold]

Please simplify this expression $\frac{n-r+1!}{n-r-1!}$ I have read the textbook and seen examples but I still do not understand.
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poles of a polynomial

What are the poles of a polynomial? Are they the same as the roots?
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1answer
23 views

polynomial factorization and equation solving

Given a polynomial equation: $$x^4+Ax^3+(B+C+D)x^2+(AB+AC)x^2+BD=0$$ where $A$, $B$, $C$, $D$ are known. Numerically I know it has complex solutions. However, I tried but failed to analytically ...
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34 views

Polynomial representation

Why is the polynomial $P(x)$ represented as $$ P(x) = a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + \cdots+ a_2 x^2 + a_1 x + a_0 \text{ ?}$$ A polynomial can be $5x^4 + 3x^3 + 7x^2 + 10x -2$ and it ...
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How do you solve part (b) to this polynomial interpolation question?

(a) Approximate $f'(x_0)$ and $f''(x_0)$ using the values $x_0-h$, $x_0$ and $x_0 + \alpha h$ $(0 < \alpha)$ by applying the polynomial interpolation method. (b) Assuming $f(x)\in C^3$, evaluate ...
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1answer
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$\forall x \,\exists k$ s.t. $f^{(k)}(x)=0$, then $f$ is a polynomial

My friend sent me the following problem: Suppose that $f$ is real analytic on $(a,b)$, and that for all $x$ in $(a,b)$ there exists a non-negative integer $k$ such that $f^{(k)}(x)=0$. Show ...
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1answer
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Product of Chebyshev polynomials of the second kind?

So Wikipedia has this formula for a product of two Chebyshev polynomials of the second kind evaluated at a fixed $x$ with different indices: $$ U_n(x)U_m(x)=\sum_{k=o}^{n}U_{m-n+2k}(x) $$ Which would ...
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1answer
48 views

A Question about the Proof of Eisenstein's Irreducibity Criterion

Statement: Let $f(x) = a_n x^n + a_{n-1} x^{n-1}+ \cdots + a_0 \in \mathbb Z[x]$. If there is a prime $p$ such that $p \nmid a_n, p \mid a_{n-1}, \dots,p \mid a_0$ and $p^2 \nmid a_0 $, then $f(x)$ ...
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47 views

primitivity of a polynomial over a field

Suppose that we have a polynomial $f(x)=ax^3-bx^2+cx-d\in\mathbb{Z}[x]$ then $f(x)$ will be called primitive if $(a,b,c,d)=1$ I have been told that over a field $F$ there is no notion of ...
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How prove that: $\left | a \right |+\left | b \right |\leq 5$ for $P(x)=x^{3}+ax^{2}+bx+1$?

Let $P(x)=x^{3}+ax^{2}+bx+1$ and $\left | P(x) \right |\leq 1$ for all x such that $\left | x \right |\leq 1$. How prove that: $\left | a \right |+\left | b \right |\leq 5$?
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Find scaling factor that minimizes f(x) - round(f(x))?

Let's say I have a function $f(x)$, which has a fractional component $\{ f(x) \} = f(x) - \lfloor f(x) \rfloor$. I would like to add a scaling factor $h(x)$, where $h(x)$ is a polynomial, such that ...
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Whether $\sum_{i=1}^k\frac{\prod_{j\neq i}(\alpha_j-\beta)}{\prod_{j\neq i}(\alpha_j-\alpha_i)}=1$ is true

Suppose we have k positive numbers: $\alpha_1, \alpha_2, ..., \alpha_k$, for any number $\beta>0$, is $$\sum_{i=1}^k\frac{\prod_{j\neq i}(\alpha_j-\beta)}{\prod_{j\neq i}(\alpha_j-\alpha_i)}=1$$ ...
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30 views

Differentiability of polynomials

Trivial question but I am confused with the notation If $p_{n-1}$ is a polynomial of degree $n-1$, is it $\in$ the differentiability class C^n$? Obviously if $p_n$ is a polynomial of degree $n$, ...
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1answer
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A question from the mod p irreducibility test's proof

Let $p$ be a prime an suppose that $f(x) \in \mathbb Z[x]$ with $\deg f(x) \geq 1$. Let $f_1(x)$ be the polynomial in $\mathbb Z_p[x]$ obtained from $f(x)$ by reducing all the coefficients of $f(x)$ ...
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1answer
14 views

Multivariate Polynomials Sage

Sorry if I'm in the wrong Stackexchange (but sage is a math program...) I'm computing something on multivariate polynomials: I have a primary variable $x$ and several other variables $a, b, c, ...
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50 views

If $a^2=b^2+c^2$ and $0<n<2$ prove $a^n<b^n+c^n$

If $a^2=b^2+c^2$ and $a,b,c$ are positive real numbers, prove (a) if $n>2$ then $a^n>b^n+c^n$, (b) if $0<n<2$ then $a^n<b^n+c^n$. Part (a) was easy to prove: $a^2=b^2+c^2$ and ...
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Find a basis and state its dimension of a $C$-vector space polynomial.

The $C$ vector space $V$ of polynomials $P(t) \in C[t]$ of degree at most $n$ and such that $P(a) = P'(a) = 0$ for $a \in C$ fixed. Indication : prove that $P(t) \in V \Leftrightarrow (t − a)^2$ ...
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Changing the order of the elements of the divided difference Polynomial Interpolation

Apparently this is rather trivial but I don't understand why what I've highlighted in green is correct.
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114 views

roots of a polynomial inside a circle

I am asked to show that for $n$ larger or equal to $2,$ the roots of $1 + z + z^{n}$ lie inside the circle $\|z\| = 1 + \frac{1}{n-1}$ Attempt1: Induction for the case $n = 2,$ the roots of $1 + z + ...
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How do you solve the coefficients of functions of two variables as part of 2nd order polynomial?

I'm having a major issue in trying to get my head around creating a formula for the graph ΔT=f(V,W) I already have two graphs for that, but they are 2D graphs. Both represent ΔT as Y but one graph ...
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1answer
31 views

Linear Equation as matrix

Using a series of 3x3 matrices multiplied together, it is possible to create a matrix which will rotate, translate, scale and invert a size 2 vector. Using a 4x4, it is possible to do this to a size ...
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1answer
30 views

Primitive-recursive functions and polynomial equations

I am looking for examples of primitive-recursive functions $f:\mathbb{N}\rightarrow\mathbb{N}$ that can not be written as a pair of polynomials, i.e. $$f(n) = m \Leftrightarrow P(n,m) = Q(n,m)$$ ...
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Is $2x^2+4$ reducible over $\mathbb C$?

I am not sure if I making some very fundamental mistake. But Gallian says that $2x^2+4$ is reducible over $\mathbb C$. If $D$ be an integral domain. A polynomial $f(x)$ from $D[x]$ is said to be ...
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How find the polynomial whose roots are $\frac{{a^2 }} {{2a^2 + bc}},\frac{{b^2 }} {{2b^2 + ca}},\frac{{c^2 }} {{2c^2 + ab}}$?

a,b,c are the roots of the polynomial $x^3 - (a + b + c)x^2 + (ab + bc + ca)x - abc$. How find the polynomial whose roots are $\frac{{a^2 }} {{2a^2 + bc}},\frac{{b^2 }} {{2b^2 + ca}},\frac{{c^2 }} ...
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About the subspace of polynomial vector space

Why the set of functions in $C\left [ 1,-1 \right ]$ such that $f\left ( -1 \right )= f\left ( 1 \right )$ is the subspace of $C\left [ 1,-1 \right ]$?
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Is there a reason for some polynomial quotients to have a remainder equals to zero?

I was helping some highschool students with factorization exercises. They had alternatives to choose the correct factor. Then one of them said to me: We use a calculator and evaluate some prime ...
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Problems from Polynomial Rings. My attempt shown.

$1.$ Let $f(x) =a_mx^m+a_{m-1}x^{m-1}+ \cdots +a_0$ and $g(x) = b_nx^n+b_{n-1}x^{n-1}+ \cdots +b_0$ belong to $\mathbb Q[x]$ and suppose that $f \circ g \in \mathbb Z[x].$ Prove that $a_ib_j$ is an ...
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Finding a polynomial's constant from its points

Let's say I was given a set of $d+1$ distinct points known to be from a polynomial $P$ of degree $d$. So: $$P = a_dx^d + a_{d-1}x^{d-1} + ... a_1x + c$$ And I have pairs $(x_i, y_i)$ such that: ...
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What basis and coordinate system is used in this quadratic Bézier triangle equation? $[x,y,z] = A*s^2 + B*t^2 + C*u^2 + D*2st + E*2tu + F*2su$

I have the following equation for a quadratic Bézier triangle, but I'm having a lot of trouble understanding how to describe it: $[x,y,z] = A*s^2 + B*t^2 + C*u^2 + D*2st + E*2tu + F*2su$ ...
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Roots Of Monic Cubic

I'm currently preparing for the USA Mathematical Talent Search competition. I've been brushing up my proof-writing skills for several weeks now, but one area that I have not been formally taught about ...
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Prove that no polynomial function represents a given sequence

I recently encountered the following question: How to find the $n$th term of the sequence $2,3,6,7,14,15,30,\dots$? I replied to that post, and gave the following answer: $S_n = ...
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Polynomial division problem

Let $f(x) = x^{10}+5x^9-8x^8+7x^7-x^6-12x^5+4x^4-8x^3+12x^2-5x-5. $ Without using long division (which would be horribly nasty!), find the remainder when $f(x)$ is divided by $x^2-1$. I'm not sure ...
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Maps preserving roots of a polynomial function over finite fields

Let $P(x_{1},\ldots,x_{n}):\mathbb{F}_{2}^{n}\rightarrow \mathbb{F}_{2}$ be a polynomial function with degree $d$ and with variables $x_{1},\ldots,x_{n} \in \mathbb{F}_{2}$. Let $S(P)=\{ ...
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1answer
23 views

Not understanding steps in Algebraic simplification

The simplification in question is that the expression goes from $(4-x)(6-x)(3-x)-8(3-x)=0$, to $(3-x)(8-x)(2-x)=0$ I don't understand how one goes from the first expression to the second. I ...
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1answer
36 views

Modulo Quadratic Polynomials

Can you, given a large number N, find a, b, c such that ax^2 + bx + c = 0 has at least N roots? All of this is in any mod you choose.
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Proof of subspace, finding a basis in polynomials

Let W = {(f(x)∈ P2[R] : f '(x) + xf(0) = 0} i) Prove that W is a subspace of P2[R]. ii) Find a basis for W. Here's what I have so far: i) I have to verify that ...
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Algebra problem about spam and polynomials

How do you know when a set of polynomials span r2 For example, P1 = x+ 5x^2+x^3 and P2 is the same. What constitutes a set to be in the span or a specific space?
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1answer
55 views

If $F$ be a field, then $F[x]$ is a principal ideal domain. Does $F$ have to be necessarily a field?

If $F$ be a field, then $F[x]$ is a principal ideal domain. Does $F$ have to be necessarily a field? My Thoughts: Suppose instead of $F$, we take the set of polynomials $R[x]$ over a commutative ring ...
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1answer
23 views

Sum of Non Real Roots of Bi Quadratic

Consider $$f(x)=8x^4-16x^3+16x^2-8x+k=0$$ where $k \in \mathbb{R}$,then find sum of non Real roots of f(x). My approach: we have $$f'(x)=32x^3-48x^2+32x-8=0$$ Also ...
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95 views

Existence of Irreducible polynomials over Z of any given degree which do not satisfy the Eisenstein's Criterion

I just came across the following interesting question which has been once discussed: Existence of Irreducible polynomials over $\mathbb{Z}$ of any given degree I was wondering if we could find such ...
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1answer
22 views

Polynomial joint pdf $f(x,y)$ such that of $f(x) \neq f(y)$

How can I build a polynomial joint pdf $f(x,y)$ for $x \in [x_1, x_2]$ and $y \in [y_1, y_2]$ such that of $f(x) \neq f(y)$ or equivalently, $x$ and $y$ are depended on each other?
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4answers
67 views

find a polynomial whose roots are inverse of squares of roots of $x^3+px+q$

Question is : Given a polynomial $f(x)=x^3+px+q\in \mathbb{Q}[x]$ find a polynomial whose roots are inverse of sqares of roots of $f(x)$ Supposing $a,b,c$ as roots of $f(x)$ we have : $a+b+c=0$ ...