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Questions tagged [polynomials]

This tag is used for both basic and advanced questions on polynomials in any number of variables, including, but not limited to solving for roots, factoring and checking for irreducibility.

4
votes
5answers
906 views

Solving quartic equations

Given the following quartic equation: $$x^4-2x^3-7x^2+8x+12=0$$ Could anyone give some techniques required to solve any quartic equation (apart from this one) if they exist?
4
votes
1answer
37 views

Galois group of $(x^3-2)(x^5-1)$ over $\mathbb{Q}$.

I am studying for my Galois theory final for tomorrow (and I'm really getting burned out), I need help with the following question: Galois group $G$ of $f(x)=(x^3-2)(x^5-1)$ over $\mathbb{Q}$. Let ...
1
vote
1answer
26 views

A question on polynomial (Divide a polynomial function)

When a polynomial $f(x)$ is divided by $(x-2),$ the remainder is $7$. When $f(x)$ is divided by $(x+1)$ the remainder is $-2$. (a) If the remainder is $px+q$ when $f(x)$ is divided by $(x-2)(x+1)$, ...
3
votes
4answers
130 views

Let $m,n\in \mathbb{Z}$ and $p(x)=x^3+mx+n$ be such that if $107\mid p(x)-p(y)\implies 107\mid x-y$. Prove that $107\mid m$.

Let $m,n\in \mathbb{Z}$ and $p(x)=x^3+mx+n$ be such that for an integers $x,y$ we have: $$107\mid p(x)-p(y)\implies 107\mid x-y$$ Prove that $107\mid m$. I'm not sure what to do here. I can only ...
1
vote
2answers
25 views

If the polynomial f(x) with real coefficients is non-negative for every $x \in \mathbb{R}$ then all its real zeros have even multiplicity

I'm dealing with this problem that says that if $f(x)$ with real coefficients is non-negative for every $x \in \mathbb{R}$ then there exists the polynomials $f_1(x)$ and $f_2(x)$ from $\mathbb{R}[x]$ ...
2
votes
6answers
97 views

Prove that $5x^2−2xy−8x+ 2y^2−2y+ 5 \ge 0$ for all $x, y\in\mathbb R$. When does equality occur?

I tried grouping the $x$'s and the $y$'s but that didn't get me anywhere. I know that $5x^2, 2y^2$, and $5$ are always positive. I am not sure what to try next.
4
votes
4answers
296 views

Sum of sixth power of roots of $x^3-x-1=0$

Question: Find the sum of sixth power of roots of the equation $x^3-x-1=0$ My First approach: Let $S_i$ denote the sum of $i^{th}$ power of roots of the given equation. Now, multiplying given ...
3
votes
4answers
3k views

How to split a quartic into two quadratics?

I have a quartic in $\Bbb Z[x]$ with very large coefficients that I know splits into two quadratics in $\Bbb Z[x]$. What is the best way to do find the quadratics?
0
votes
1answer
32 views

Assumption $d>2$ on Proposition 2.12 from Knapp's Elliptic Curves

I'm going through Knapp's book on elliptic curves and I got stuck in a minor detail. This is a part of the proof of Proposition 2.12: I could understand everything except for this little detail: ...
2
votes
0answers
29 views

Determine derivative of polynomial from graph on a multiple choice question [on hold]

On a multiple choice question I need to determine which of the following graphs: Multiple choice graphs When derivated, yields: The graph Is there a general, fastest way to solve this kind of ...
0
votes
2answers
36 views

Is this proof of a irreducibility criterion in an integral domain correct?

This is an exercise from Grillet's "Abstract Algebra" (page $145$, proposition $10.10$). Let $R$ be an integral domain, let $I$ be an ideal of $R$, and let $\pi\colon R\to R/I$ be a canonical ...
0
votes
1answer
20 views

What are the origins of the Routh-Hurwitz Criterion?

The Routh-Hurwitz criterion and method are usually taught in a cookbook format. Essentially, you follow a recipe for placing the coefficients into a table and perform "figure 8" multiplication and ...
0
votes
1answer
56 views

If $f(x)$ is a polynomial, it cannot represent primes only [on hold]

If I take $u = f(x)$ and $v = f(x + ku)$ where $k$ is any integer and then if somehow I can prove that $u$ is a factor of v but I don't have any idea about the next step.
0
votes
1answer
25 views

Can roots be $y$-intercepts for the quadratic function

I know this must be stupid question, but I was wondering why cannot a quadratic or any polynomial equation be in format of $$x=ay^2+by+c$$ and to find roots we set $x=0$. In short, can the $y$ ...
0
votes
0answers
26 views

Find basis to sub vector spaces

V = $\Bbb R_3 [x]$ and W,U $\subseteq$ V are sub vector spaces. U=$span${$1 - x, x^2, x^2-x^3, -1+x-x^2+2x^3$} W={p(x)$\in\Bbb R_3 [x]$ | p(1)=0 ^ p(2)+p(0)=0} Find basis to W, U+W, U$\cap$W
2
votes
1answer
365 views

$f$ irreducible over $\mathbb{Z}_{p}$ implies $f$ is irreducible over $\mathbb{Q}$

Let $f \in \mathbb{Z}[x]$ be a non-constant polynomial and let $p$ be a prime number which is not a divisor of the leading coefficient of $f$. I need to prove that if $f$ is irreducible over $\mathbb{...
9
votes
1answer
3k views

Quick ways to _verify_ determinant, minimal polynomial, characteristic polynomial, eigenvalues, eigenvectors …

What are easy and quick ways to verify determinant, minimal polynomial, characteristic polynomial, eigenvalues, eigenvectors after calculating them? So if I calculated determinant, minimal polynomial,...
6
votes
2answers
371 views

Compute the $m$th derivative of the sigmoid function $x(t)=1/(1+e^{-at})$

Compute the $m$th derivative of the sigmoid function $x(t)=1/(1+e^{-at})$. I hardly understand English. I wrote a question using translation. I want to generalize. $a$ is a constant $$x(t) = \frac{...
11
votes
7answers
118 views

Solve $\sqrt{1 + \sqrt{1-x^{2}}}\left(\sqrt{(1+x)^{3}} + \sqrt{(1-x)^{3}} \right) = 2 + \sqrt{1-x^{2}} $

Solve $$\sqrt{1 + \sqrt{1-x^{2}}}\left(\sqrt{(1+x)^{3}} + \sqrt{(1-x)^{3}} \right) = 2 + \sqrt{1-x^{2}} $$ My attempt: Let $A = \sqrt{1+x}, B = \sqrt{1-x}$ and then by squaring the problematic ...
0
votes
0answers
26 views

The polynomial $aX^2+bX+c$ of degree $2$ in a field $K$ with char$(K)\neq 2$ is irreducible if and only if $b^2-4ac$ is not a square in $K$.

I have to prove the following: The polynomial $aX^2+bX+c$ of degree $2$ in a field $K$ with char$(K)\neq 2$ is irreducible if and only if $b^2-4ac$ is not a square in $K$. I have never worked with ...
1
vote
0answers
30 views

Polynomials in the Pancake problem

I noticed something interesting in this table. The columns can be expressed by polynomials of degree k. I toke the first $k+1$ numbers from each column and used Lagrange's interpolation. Surprisingly, ...
-4
votes
0answers
18 views

The partial graph of $y=f(x)$ is shown below.Provide an accurate sketch of graph $y=0.5f(0.5x)$ [on hold]

The partial graph of $y=f(x)$ is shown below. Provide an accurate sketch of graph $y=0.5f(0.5x)$
0
votes
1answer
39 views

The partial graph of $y=f(x)$ is shown below, provide an accurate sketch of the graph $y-2=f(x+3)$ [on hold]

The partial graph of $y=f(x)$ is shown below, provide an accurate sketch of the graph $y-2=f(x+3)$ Please if somebody have idea,thank you
4
votes
2answers
1k views

Minimum value of The polynomial

What is the minimum value of the expression given below? $\ x^8-8x^6+19x^4-12x^3+14x^2-8x+9$ Now to solve this I have resolved the expression, like following, $\ (x^2+2x)^2.(x^2-2x)^2+3.(x^2-2x)^2+2(...
16
votes
3answers
5k views

Proving that a polynomial is not solvable by radicals.

I'm trying to prove that the following polynomial is not solvable by radicals: $$p(x) = x^5 - 4x + 2 $$ First, by Eisenstein is irreducible. (It is not difficult to see that this polynomial has ...
5
votes
4answers
53 views

Factoring $(x+a)(x+b)(x+c)+(b+c)(c+a)(a+b)$ and use the result to solve an equation

I managed to prove that $(x+a+b+c)$ is a factor of $$(x+a)(x+b)(x+c)+(b+c)(c+a)(a+b)$$ Then I was asked to use the result to solve $$(x+2)(x-3)(x-1)+4=0$$ I know by comparison, $a=2, b=-3, c=-1$, ...
-1
votes
0answers
25 views

Find a basis and coordinates for a second degree polynomial

Find a basis $B$ for $P_2$ $[p]_B = \begin{bmatrix}p(0)\\p(1)\\p(2)\end{bmatrix}$ and its coordinates to a second degree polynomial The solutions says: $p(x) = p(0)e_1(x) + p(1)e_2(x) + p(2)e_3(x)...
0
votes
1answer
29 views

How to solve this 4x4 equation system for a cubic spline?

I am attempting to create a simple cubic spline between these lines: I have worked out the four equations as: (1) $1 = An^3 + Bn^2 + Cn + D$ (2) $g^{m-t} = At^3 + Bt^2 + Ct + D$ (3) $0 = 3An^2 + 2Bn ...
63
votes
3answers
6k views

Polynomials irreducible over $\mathbb{Q}$ but reducible over $\mathbb{F}_p$ for every prime $p$

Let $f(x) \in \mathbb{Z}[x]$. If we reduce the coefficents of $f(x)$ modulo $p$, where $p$ is prime, we get a polynomial $f^*(x) \in \mathbb{F}_p[x]$. Then if $f^*(x)$ is irreducible and has the same ...
0
votes
0answers
10 views

How do i show working to Derive a quadratic Bezier curve [on hold]

SO the formula for a Quadratic bezier curve is: P(t)=(1−t)3P0+3t(1−t)2P1+3t2(1−t)P2+t3P3 But the quesiton is how to derive the formula Please help
1
vote
2answers
41 views

How to compute gcd of two polynomials efficiently

I have two polynomials $A=x^4+x^2+1$ And $B=x^4-x^2-2x-1$ I need to compute the gcd of $A$ and $B$ but when I do the regular Euclidean way I get fractions and it gets confusing, are you somehow able ...
2
votes
1answer
38 views

Polynomial which fix set of integers with only 1 in base 10 [duplicate]

Let be $A$ the set of integers non-negative which only have 1 in their base 10 expansion (e.g. $111111$, $11$, $1$). I would like to find all polynomials $P \in \mathbb{C}[X]$ such that $P(A) \subset ...
3
votes
0answers
123 views

If $x^n-3x^{n-1}+2x+1=0$, compute $\sum_{k=1}^n \frac{x_k}{x_k-1}$ [closed]

Let $x_1, x_2, \ldots, x_n$ be the $n$ roots of the equation $x^n-3x^{n-1}+2x+1=0$. Calculate the sum $\sum_{k=1}^n \frac{x_k}{x_k-1}$. Can somebody give me some tips, please? It's pretty hard to ...
0
votes
1answer
8 views

Is a function of admissible heuristics in A* search admissible?

I don’t understand how to approach this problem. $h_1, h_2, h_3$ are three admissible heuristics for an optimisation problem to be solved using A* search. Is the heuristic defined by $$h(n) = \frac{...
0
votes
1answer
33 views

Grobner basis sufficient condition

Assume $f \in K[x_1,\dots ,x_m],x_i\in F$ $$p \in K[x_1,\dots, x_m], p = \sum_{i}\phi_ix_1^{{a}_{i1}}x_2^{a_{i2}}\dots x_n^{a_{in}}, L(p) := \phi _lx^l : x^a \prec x^l \forall a\in A(p) \...
0
votes
0answers
13 views

The smoothing property of Bernstein polynomial

The smoothing property of Bernstein Polynomial, proved by Kelisky and Rivilin in 1967, $$ \lim_{k\rightarrow\infty}B\;^{(k)}\left(f;x\right)=\left(1-x\right)f\left(0\right)+xf\left(1\right) $$ can be ...
0
votes
0answers
82 views

existence and uniqueness of solutions to a system of polynomial equations on positive reals

The following conjectures have come up in a project studying mechanisms to enhance incentives to contribute to public goods: There is a set I of charities. Initially, each $i\in I$ has an amount $a_{...
0
votes
1answer
15 views

Calculate a Primitive Polynomial LFSR

I tried to search on the internet, to read my course multiple times, but the only thing I see are definitions of the primitive polynomials for an LFSR. I have an exercise: Find the primitive ...
1
vote
2answers
38 views

Is there a sum of “n+j” terms equivalent to n!

Is there any function so that... $$\sum_{k=0}^{n} f(k) = n!$$ or, $$\sum_{k=0}^{n+j} f(k) = n!$$ where j is any arbitrary integer?
0
votes
2answers
51 views

Solving a polynomial by grouping and factoring - why does this answer have $\pm3i$?

I am asked to solve for x in the polynomial using factoring and grouping: $5X^3+45X=2X^2+18$ My working: $5X^3-2X^2+45X-18$ $X^2(5X-2)+9(5X-2)$ $(X^2+9)(5X-2)$ So: $X^2+9=0$ $X^2=-9$ $X=i\sqrt{...
0
votes
0answers
31 views

How do i derive the equation of a Bezier curve?

I have encounted a problem so i am trying to create and plot a Bezier curve and i have four control points. I have to link the application of polynomials and Pascals triangle within the answer. Now i ...
1
vote
1answer
36 views

Non zerodivisors in ideals of polynomial rings [duplicate]

The question is the following: Let $f$ be a polynomial of the ring $R[x_1, \ldots, x_n]$, with $R$ any ring, and let $\mathrm{cont}(f)$ be the ideal generated by the coefficients of $f$. Why if $\...
5
votes
1answer
56 views

Is it possible to graph complex zeros of a polynomial?

I am sorry if this question is a complete nonsense, but keep in mind that I am a senior in high school, so my math knowledge is really low. My question is, can you graph complex zeroes on a three ...
1
vote
2answers
25 views

GCD of $f=X^3 +9X^2 +10X +3$ and $g= X^2 -X -2$ in $\mathbb{Z}/5\mathbb{Z}$.

I have to calculate the gcd of $f=X^3 +9X^2 +10X +3$ and $g= X^2 -X -2$ in $\mathbb{Q}[X]$ and $\mathbb{Z}/5\mathbb{Z}$. In $\mathbb{Q}[X]$ I got that $X+1$ is a gcd and therefore $r(X+1)$ since $\...
1
vote
1answer
42 views

Irreducible polynomials for that each output is divisible by an integer n

Feel free to delete this question if it has been asked somewhere else before. I've recently stumbled upon this question on the Mathematics StackExchange and I've wondered how the polynomials for ...
2
votes
0answers
16 views

Tough test polynomials for (finite precision) complex root finding methods, especially Aberth's method

Today I have implemented Aberth's method for complex polynomial root finding. And I have to say I am enchanted about its astonishing performance and its intriguing simplicity. Before I go on believing ...
0
votes
0answers
16 views

algebraic field extensions and polynomials

Let $K$ be a field and $f$ be a polynomial, whose coefficients are algebraic over $K$. If we have a factorization $f=gh$ and the highest coefficient of $g$ is algebraic over $K$, is it true that the ...
-1
votes
1answer
51 views

Show that $\frac{x(x-1) \dots (x-n+1)}{n!} \in \mathbb{Z}$ with $x \in \mathbb{Z}$ [duplicate]

Problem: Let polynomial $Q_n (x) = \frac{x(x-1) \dots (x-n+1)}{n!} \in R[x]$ for some ring $R$. Show that $\forall t \in \mathbb{Z}, Q_n (t) \in \mathbb{Z}$. My solution: For each $t \in \mathbb{Z}$, ...
2
votes
4answers
46 views

Long division of $\frac{3x^3-x^2-13x-13}{x^2-x-6}$

I'm self-studying from Stroud & Booth's amazing textbook "Engineering Mathematics", and am on the "Partial Fractions" chapter. As part of an exercise I need to do long division of two polynomial ...
0
votes
0answers
16 views

How to analytically calculate a specific pseudoinverse for multinomial root finding

Consider the following problem: $$d_1=a_{1,1}x+a_{1,2}y+a_{1,3}x^2+a_{1,4}2xy+a_{1,5}y^2$$ $$d_2=a_{2,1}x+a_{2,2}y+a_{2,3}x^2+a_{2,4}2xy+a_{2,5}y^2$$ and suppose we can represent it as follows: $$\...