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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Example such that $ f_1(x)$ is reducible but $f(x)$ is irreducible.

(The (mod p) Irreducibility Test) 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 ...
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362 views

Only 12 polynomials exist with given properties

Prove that there are only 12 polynomials that have all real roots and whose coefficients are $-1$ or $1$. Zero coefficients are not allowed, and constant polynomials do not count. Two of them ...
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3answers
54 views

Prove that for any real number the inequality is true: $x^4-x^3+5x^2 > 3x - 6$

Prove that for any real number the inequality is true: $x^4-x^3+5x^2 > 3x - 6$ The only way I could do this is to transform this inequality to: $x^4-x^3+5x^2-3x+6> 0$ and then sketch the ...
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“completing the cube”

Given a quadratic polynomial $p(z) = z^2 +a z +b$ in $\mathbb{C}[x]$, we can "complete the square" to write $p(z) = g(f(z)^2)$ where $f,g$ are translations. \begin{align*} f(z) = z + a/2 && ...
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How do I solve massive system of equations (with lots of variables) quickly?

Just wondering how to solve system of equations involving 3+ unknowns quickly. In my math class, we're given questions like these which involve solving huge system of equations on a time limit, ...
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Integral Inequality $\leq n^{3/2}\pi$

$ p(x)\in\mathbb{R[X]} $ is a polynomial of degree $n$ with no real roots. Show that: $ \int\limits_{-\infty}^{+\infty}\dfrac{(p'(x))^2}{(p'(x))^2+(p(x))^2}\,dx \leq n^{3/2}\pi.$ It's easy to see ...
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1answer
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Newton's sum help

How would one solve the following system with Newton's sums and Vieta's relations?: $$x+y+z=14$$ $$x^2+y^2+z^2=14$$ $$x^3+y^3+z^3=34$$ I have taken an algebra lesson in the awesome math academy, but ...
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5answers
178 views

Curious Binomial Coefficient Identity

Consider the following set of identities: ${m+1\choose 1}={m\choose 1}+1$, ${m+1\choose 2}=2\binom m 2 - {m-1\choose 2}+1$, ${m+1\choose 3}=3\binom m3-3{m-1\choose 3}+{m-2\choose 3}+1$, ... This set ...
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What is that function? Polynomial?

Is it a polynomial or rational polynomial or else? $y = \dfrac{a}{x^4} + \dfrac {b}{x^2} + c$ I need to fit a curve to a discrete data of that form, so I need to know what fitting to use.
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Proving a version of maximum modulus principle elementarly.

There is this version of maximum modulus principle: If $P$ is a non-constant polynomial, then $|P|$ doesn't have a local maximum. I know that if $P$ is non-constant, then $|P(z)| ...
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Find the coefficient of $x^{30}$.

Find the coefficient of $x^{30}$ in the given polynomial $$ \left(1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^{10}+x^{11}+x^{12}\right)^5 $$ I don't know how to solve problems with such high degree.
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1answer
53 views

Is there a formula for the closed form for $ \displaystyle \sum_{r=1}^\infty \frac{\sum_{k=1}^r k^n}{r!}$ for any positive integer $n$?

Is there a formula for the closed form for $ \displaystyle \sum_{r=1}^\infty \frac{\sum_{k=1}^r k^n}{r!}$ for any positive integer $n$? I tried Faulhaber's formula and Bell number but couldn't ...
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1answer
43 views

An approach to proving that $\Bbb{Q}[x,y]/(x^3-y^2)$ is isomorphic to $\Bbb{Q}[t^2,t^3]$

I have to prove that $\Bbb{Q}[x,y]/(x^3-y^2)$ is isomorphic to $\Bbb{Q}[t^2,t^3]$. My approach: Let us consider $t^2$ and $t^3$ as separate variables $x$ and $y$. The relations that hold for them ...
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1answer
36 views

general theorem on roots of a polynomial needed to show it's identically zero.

Polynomial degree k, one variable, if it's zero at k+1 values, then it's identically zero. Can someone point me to a proof of this? I know derivatives at points can count as these roots (if k-degree ...
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coefficients of polynomial and binomial expressions

Let us say we are given a polynomial p(x)=$\sum_k a_k x^k$. In order to find $\sum_k a_k$ we simply need to evaluate p(1), and similarly there are many other tricks. Is there any trick to evaluate ...
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47 views

Prove $\lim_\limits{x\to\infty}\dfrac{P_k(x)}{P_{k+1}(x)}=0$ [on hold]

Prove $\lim_\limits{x\to\infty}\dfrac{P_k(x)}{P_{k+1}(x)}=0$ by limits. $P_k(x)$ is defined as a polynomial of degree $k$.
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1answer
318 views

Algorithm to find the maximum/minimum of a polynomial without graphing.

For a quadratic equation of the form $y=ax^2+bx+c$ the max/min occurs at $x=-\frac{b}{2a}$. Is there any hard and fast equation like this for polynomials of degree $\geq 4$?. For such polynomials the ...
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1answer
37 views

Equation that defines multi-dimensional polynomial

In two-dimensions a complete n-th degree polynomial is given by $P_n(x,y) = \sum_{k=0}^{n}\alpha_kx^iy^j \qquad i+j \leq k \qquad (1)$ . However, now I am dealing with the following two-dimensional ...
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Prove: if $a$ and $b$ are algebraic, then $a + b$, $a - b$ and ab are also algebraic

I have to prove the following: If $a, b \in \mathbb{C}$ and are both algebraic over $\mathbb{Z}$, then: $a + b$ is algebraic over $\mathbb{Z}$ $a - b$ is algebraic over $\mathbb{Z}$ $ab$ is ...
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Cardinomials: Like cardinalities, but polynomial valued

I want to see if this notion is known (or if it makes sense). Let $F$ be a field. Let $A$ be a finite dimensional commutative unital algebra over $F$. Let $X_1$, $X_2 \in A$ etc. be such that their ...
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3answers
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Consider the equation [x^5+x=10] show that i)the equation has only 1 real root ii)this root lies between 1 & 2 iii)the root must be irrational [on hold]

This equation obviously has 5 roots..If they are considered as a,b,c,d,e then a+b+c+d+e=0,abcde=-10..but what next?If i procced through contradiction will it help?
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1answer
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Find $n$ such that $x^2 + x + 1$ is a factor of $(x+1)^n - x^n - 1$.

I have to find the form of n i.e. whether n is even or odd and whether it is multiple of 2 or 3 such that: $x^2 + x + 1$ is a factor of $(x+1)^n - x^n - 1$. What I tried: $x^2 + x + 1 = (x + 1)^2 - ...
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0answers
27 views

Given $f=x^4+x+1 \in \mathbb Z_{2}[x]$ is primitive, write down an $m$-sequence ${a_n}$ associated to $f$

I'm not sure how to solve this question exactly. I know that the period will be 15 but I don't know how to construct the $m$-sequence.
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2answers
486 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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0answers
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Solving quartic equation? (Cardano/Ferrari)

today I've written a little Code-Snippet that is based upon the steps that are mentionned in this wikipedia-Article to solve a general quartic polynom. Here's my matlab-implementation: ...
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Is the given binomial sum almost everywhere negative as $K\to\infty$?

The binomial sum is as follows: $$\mathcal {L}^K(\theta)= \sum_{i=\lceil{K/2}\rceil}^K \binom{K}{i}\theta^i\left((1-\theta)^{K-i}-\frac{1}{2}(1-\theta)^{-K}(1-2\theta)^{K-i}\right)$$ It can be found ...
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5answers
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Cubic Equation. (Factorisation)

I'm given this question, factorise $4x^3-7x-3$. Is this answer acceptable? $(x+\frac{1}{2})(x-\frac{3}{2})(x+1)$.
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1answer
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+50

Differentiate polynomials in $\mathbb{Z}_2[x]$

It seems suggested that the differential of a polynomial in $\mathbb{Z}_2$ is as I would expect: $$\begin{align} &f = x^6 + x^3 + x + 1 \\ &f' = 6x^5 + 3x^2 +1 \mod 2 \\ &f'= x^2 + 1 \\ ...
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1answer
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Solving the Cubic Equation (using Lagrange Resolvents)

This is from my textbook. I am having trouble working out the calculations that the author skips over. So we start with the polynomial $\ X^3 - aX^2 + bX -c$ with zeros $x_1,x_2,x_3$. Then we define ...
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1answer
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How to find representation of polynomial w.r.t different basis

Let $B$ be the basis of the vector space of polynomials of degree less than or equal to 2. $B = \{1, t-1,(t-1)^2\}$. Let $u = 2t^2-5t+6$. How do you find $u_b$, the coordinate vector of $u$ relative ...
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Solving a Quintic polynomial

The polynomial $F(x) = x^5-9x^4+24x^3-24x^2+23x-15$ has roots $x=1$ and $x=j$. Calculate all the roots of the polynomial. I was told I had to use radicals or similar to solve this but after reading ...
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+100

Help me understand Gröbner basis result please

I'm practicing a bit with Gröbner bases but I'm not understanding the following result I obtain from Mathematica: ...
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1answer
66 views

Ordered Pairs of Polynomials

Professor proposed this problem to the class today. Suppose we had $P_1(x), P_2(x) \in \mathbb{Z[x]}$, $n, a \in \mathbb{Z}$. How many ordered pairs exist such that ...
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$S_k(x+y)-S_k(x)-S_k(y)$ where $S_k$ is symmetric polynomial

Let $S_k$ be the $k$-th symmetric polynomial of $n$-variable. How can I rewrite $$S_k(x+y)-S_k(x)-S_k(y)$$ by just using $x,y,S_1,S_2,\cdots S_{k-1}$ where $x=(x_1,x_2,\cdots,x_n)$ and ...
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Show that $P$ is divided to simple roots knowing that $a_{k}^2-4a_{k-1}a_{k+1}>0$

Let $P(X)=a_0+a_1X+..+a_nX^n\in R[X]$ Assume that $\forall k, a_k>0$ and $a_{k}^2-4a_{k-1}a_{k+1}>0$ Show that $P$ is divided to simple roots in $R[X]$. i.e. ...
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1answer
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if $f(n+1)-f(n)=P(n)$, exist a polynomial $Q(x)$ such that for all $n \in \mathbb{Z}$ : $Q(n)=f(n)$

Let $f:\mathbb{Z} \to \mathbb{Z}$ such that, exist a polynomial $P(x)$: $$f(n+1)-f(n)=P(n)$$ for all $n \in \mathbb{Z}$ Prove that exist a polynomial $Q(x)$ such that for all $n \in ...
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seeking for Newton's like inequalities as sufficient condition for polynomial to have only real zeros

For polynomial $P_n(x)=\sum_{k=0}^n a_k x^k, a_k>0$, it is known that a necessary condition for $P_n(x)$ to have only real zeros is that Newton's inequality holds: ...
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Recursive relationship for Peano Baker Series

The Peano Baker Series is a integral has the following form $$\varPhi(h,0)=I+\intop_0^h G(t_{1}) \, dt_1 + \intop_0^h G(t_1) \intop_0^{t_{1}} G(t_2) \, dt_2 \, dt_1 + \intop_0^h G(t_1) ...
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Series Expansion from Polynomial w/ Coefficients [on hold]

I have four coefficients to a 4-the order polynomial. Besides having some stroke of luck finding a pattern (that would be difficult considering the coefficient values) what is the best way to approach ...
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1answer
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An obstacle encountered in a proof of the existence of a best approximating polynomial of degree $\leq n$

Let $n \in \{0, 1, 2, \dots\}$, let $a, b \in \mathbb{R}$ be such that $a < b$ and let $f \in \mathcal{C}[a, b]$ be a real function that is continuous on the non-degenerate, compact interval $[a, ...
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1answer
49 views

Taylor polynomial for an integral

This is the first time encountering a Taylor expansion along with an integral, so I am wondering how I should proceed. Question: $Consider \space the \space function$ $$F(x) = ...
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1answer
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I have to show $p=p(x-\lambda)$ if and only if they have the same zeros in $F$

Suppose $F$ is a field, $|F|\geq n \geq 2$. Given $\lambda \in F$ I know $p,p(x-\lambda)\in F[x]$ are irreducible monic polynomials. I have to show $p=p(x-\lambda)$ if and only if they have the same ...
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Is the convolution of a function $f(x)$ and a polynomial $p(x)$ always a polynomial?

After reading the following question: How do I prove a convolution is a polynomial? I want to ask if that is always the expected result, that is to say, does the following holds? A convolution ...
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878 views

Linear Algebra: Dual Basis Problem

Problem Let $V$ be the vector space of all polynomial functions $p$ from $\mathbb{R}$ to $\mathbb{R}$ which have degree two or less. Define three linear functionals on $V$ by ...
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Zero Homogeneous Polynomials

Let $f:\mathbb{R}^n\rightarrow \mathbb{R}$ be a homogeneous polynomial of degree n. Is it true that if $\forall x,f(x)=0$, then the coefficients of $f$ are all zero?
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Writing a series of polynomial equations of certain degree from a sequence of binary bits using Magma

How do I write a series of polynomial equations of a specified degree from a sequence of binary bits using Magma. So far, I have the following code for converting a decimal sequence to binary. ...
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1answer
31 views

Find all polynomials such that $P(A)\subset U$ for a countable subset of the unit circle $U$

I recently answered a question, in which I proved that If a polynomial fixes the unit circle then $P$ is a monomial (a classical result),i,e: $$\forall P\in \Bbb C[X]\ \ \ \ (\forall z\in \Bbb C \ \ ...
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1answer
19 views

Write down basis for the set of all polynomials $f(x)$ of degree at most 5 such that $f(2) = 0$.

Write down basis for the set of all polynomials $f(x)$ of degree at most 5 such that $f(2) = 0$. I know there are lots of answers you could write, but would this be correct: $\{(x-2)^5, (x-2)^4, ...
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0answers
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Polynomial Division for crc

I did this question by just using the xor long division of the binary, but my teacher said he doesn't want it done that way, but want me to use polynomial Division. I have no clue how to do this, and ...
2
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1answer
30 views

If $|\det(A+zB)|=1$ for any $z\in \mathbb{C}$ such that $|z|=1$, then $A^n=O_n$.

Let $A,B\in \mathcal{M}_n(\mathbb{C})$ such that $AB=BA$ and $\det > B\neq 0$. a) If $|\det(A+zB)|=1$ for any $z\in \mathbb{C}$ such that $|z|=1$, then $A^n=O_n$. b) Is the ...