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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.

0
votes
1answer
21 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
14 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: ...
0
votes
0answers
24 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
3
votes
3answers
78 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 ...
11
votes
7answers
104 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
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0answers
24 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 ...
-4
votes
0answers
17 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
37 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
5
votes
4answers
51 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
24 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
27 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 ...
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
34 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 ...
2
votes
6answers
94 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.
0
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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
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
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 ...
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) \...
1
vote
2answers
36 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
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 ...
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{...
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 $\...
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 ...
4
votes
1answer
54 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 ...
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 ...
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, ...
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
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 $\...
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
15 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: $$\...
-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
2answers
59 views

If $F\subseteq K$ are fields, $\alpha \in K$ Prove $\alpha$ is algebraic over $F$ [on hold]

If $F\subseteq K$ are fields, $\alpha \in K$, and $K$ is an extension field of $F$. Prove the following are equivalent: $\alpha$ is algebraic over $F$ $F(\alpha)=F[\alpha]$ $|F(\alpha):F|$ is ...
2
votes
1answer
33 views

Show that any complex polynomial of degree $n \ge 2$ has finitely many neutral or attracting orbits

Let $f: \mathbb{C} \to \mathbb{C}$ be a complex polynomial of degree $n \ge 2$. A point $w \in \mathbb{C}$ is a periodic point of $f$ of minimal period $p$ if $f^{\circ p}(w) =w$ and $p$ is the ...
0
votes
1answer
29 views

For complex polynomial $f(z)$ show there are finitely many $w$ such that $f(v) = w \neq v$ implies $f'(v) =0$.

Let $f(z) : \mathbb{C} \to \mathbb{C}$ be a complex polynomial of degree $n \ge 2$. Consider the set $$A = \{w \in \mathbb{C} : \text{there exists } v \neq w \text{ with } f(v) = w \text{ and } f'(v)...
0
votes
1answer
29 views

Non-isomorphic graphs with same Tutte polynomial

I've been looking for some non-isomorphic graphs with the same Tutte polynomial. I'm aware of this thread and this thread, however my understanding of matroids is non-existent, and they are a bit ...
1
vote
6answers
95 views

real solution of equation $(x^2+6x+7)^2+6(x^2+6x+7)+7=x$ is

Number of real solution of equation $(x^2+6x+7)^2+6(x^2+6x+7)+7=x$ is Plan Put $x^2+6x+7=f(x)$. Then i have $f(f(x))=x$ For $f(x)=x$ $x^2+5x+7=0$ no real value of $x$ For $f(x)=-x$ $x^2+8x+7=0$...
0
votes
2answers
70 views

Find the value of $\alpha^{\frac13}+\beta^{\frac13}$

If $f(x)=x^2-5x+8, f(\alpha)=0$ and $f(\beta)=0$ then find the value of $\alpha^{\frac13}+\beta^{\frac13}$ $$\alpha+\beta=5$$ $$\alpha \beta=8$$ $$\alpha^{\frac 1 3}=\frac 2 {\beta^{\frac 1 3}}$$ ...
0
votes
1answer
21 views

$f(x)=(x-a)(x-a_2)…(x-a_n)\in F[x]$ where $F$ is a field and $a_j\in $ for $j=1,2,…,n$ has no repeated roots iff gcd$(f(x),f'(x))=1\in F[x]$

This makes sense to me if $a_j\ne a_k$ for $j\ne k$ as $(x-a_j)=0 \implies a_j$ is a root of $f(x)$. So if all $a_j$ are different, then all the roots will be different. Do I have to somehow show this ...
2
votes
3answers
32 views

Prove The Derivative Rules in the Ring of Polynomials

Let R be a commutative ring with unity element 1. Let $f(x)\in R[x]$ and define its derivative as $f'(x)=r_1 +2(r_2)x+...+n(r_n)x^{n-1}$. Prove that $(f+g)'(x)=f'(x)+g'(x)$ and that $(fg)'(x)=f'(x)g(x)...
1
vote
0answers
35 views

When does $\sum_{k=0}^M a_k x^{2k}$, $a_k \in \mathbb{R}$ have no roots in $[0,1]$?

When does $\sum_{k=0}^M a_k x^{2k}$, $a_k \in \mathbb{R}$, $M \in \mathbb{N}$ have no roots in $[0,1]$? There is nothing special about the $1$, the question can be generalized to $[0,c]$ but that ...
2
votes
1answer
38 views

A polynomial whose roots form an arithmetic progression

Let $f$ be a fourth degree polynomial whose roots form an arithmetic progression. Prove that $f'$'s roots also form an arithmetic progression. I didn' t make much progress, I just wrote $f(x) =a(x-b-r)...
0
votes
0answers
16 views

Coloring of hypergraph with polynomial implication

Sorry to bother again with my misunderstandings, but I encountered yet again an issue with the topic of coloring in graphs and again I require some help to deal with this exercise: Let $\mathbb{F}$ ...
0
votes
0answers
32 views

Dealing with a polynomial sum

I am trying to approximate the following function, $$f(x)=\frac{\sum^{N}_{c=1}\gamma_c x^{c+k-2}-\sum^{N}_{c=1}\beta_c x^{c-1}}{\sum^{N}_{c=1}\alpha_c x^{c}}$$ where $\alpha_c$'s, $\beta_c$'s and $\...
0
votes
5answers
24 views

Which of these collections of elements span $P_2(\mathbb{R})$

Consider the $\mathbb{R}$-vector space $P_3(\mathbb{R})$ of real polynomials $$a_0+a_1X+a_2X^2,$$ of degree $\leq 2$. Which of the following collections of elements span $P_3(\mathbb{R})$? $1,X,X^2$ $...
0
votes
1answer
49 views

How do I produce a Bézier curve with 4 points and without just using a website?

I have these for points, (1,-2), (4,3), (12,3) and (15,-3), and I was wondering how do I make a model out of this, and how would this relate to polynomials and Pascal's triangle? I have this formula: ...
0
votes
5answers
43 views

Proof strategy polynomial division

I'm given the following polynomial $$ p(x) = x^4 -2x^3 +6x^2-10x+5 \in \mathbb{Q}[x] $$ I need to prove that $(x-1)^2|p(x)$. Which are the possible ways to prove that? One way is to do the ...
1
vote
0answers
24 views

Number of distinct roots between complex roots among three polynomials

I want to prove that any three relatively prime polynomials $A, B, C \in \mathbb C[X]$ verifying $A+B+C=0$ have at least $1+\max(\deg A, \deg B, \deg C)$ distinct roots in total among each other. I ...
0
votes
0answers
28 views

Where can I study polynomials online?

I have one good book on polynomials. It is called (Level of Knowledge of Polynomials) by its author S. L. Tabachnikov. In principle, I could study it, but I am still mastering the Agebra I. Not long ...
0
votes
0answers
11 views

Set of roots Hermite polynomials(probabilistic type)

What is the nature of the set of all roots of the Hermite polynomials? They’re known to be all real. Is it a dense set? If not, what are the limit points?Are the limit points also roots? Are the limit ...