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.

2
votes
3answers
44 views

Determine polynomial function of degree 4.

The graph of a polynomial function $f(x)$ of degree $4$ with real coefficients has a local maximum at $(-3|3)$ and a local minimum at $(1|0)$ and no other local extremal points. Determine the function....
1
vote
1answer
16 views

Order Polynomial of Matrix Commutative Property

Given a matrix $A$ and a polynomial $P$ as its linear factors $$\prod_{i=1}^{n}(\lambda_i-x)$$ For $P(A)$, does it matter in which order the linear factors are multiplied, seeing as matrix ...
1
vote
1answer
23 views

Kernel of endomorphism for polynomials from $f(x)$ to $f(2x+1)$

I am looking for a kernel of a map, producing $f(2x+1)$ out of $f(x)$. where $f(x)$ is an arbitrary polynomial of degree $n$. I thought of trying to write this transformation as a matrix and then to ...
1
vote
0answers
19 views

Why is the sign of the last term in a Sturm sequence always constant?

I have been given the definition of a Sturm sequence ($p_1,\dots, p_m$) as follows: $$p_0 = a \in \mathbb{Q}[x]\\ p_1 = a'\\ \forall 1 < i < m: p_i(t) = 0 \implies \text{sig}(p_{i-1}(t))=-\text{...
1
vote
1answer
78 views

what is the root of this polynomial?

Let $f_n(x)=\prod\limits_{i=1}^n (x+i)-n!=(x+1)(x+2)\cdots(x+n)-n!$ $n$ is a positive integer. What are the roots of the polynomial for a given $n$ except $0$? Or determine the real part of the ...
3
votes
0answers
54 views

Prove that $\mathbb{Z}[x]/(x^2 - 3) \cong \mathbb{Z}[\sqrt{3}]$.

Here is the statement of the problem: Let $(x^2 - 3)$ be the ideal of $\mathbb{Z}[x]$ generated by $x^2 - 3$ and let $\mathbb{Z}[\sqrt{3}] = \{a + b\sqrt{3} : a,b \in > \mathbb{Z}\}$. Prove ...
-1
votes
2answers
61 views

I am stuck on this one question where I have to find the coefficient of x^2. It is non-calculator. [closed]

What is the coefficient of $x^2$ in $$ \left(4-x^2\right)\left[\left(1+2x+3x^2\right)^6-\left(1+4x^3\right)^5\right] $$ A calculator cannot be used.
3
votes
1answer
68 views

Can we let $n$ go to $\infty$ after applying the fundamental theorem of algebra to a polynomial of degree $n$?

Recently I came across a proof of the infinite product for $\sin z$ (https://www.sciencedirect.com/science/article/pii/0022247X77902347). It applies the fundamental theorem of algebra to $$p_{n}(z)=\...
0
votes
2answers
42 views

Prove that polynomial doesn't admit a particular solution

Prove that the equation$$x^4+(a-2)x^3+(a^2-2a+4)x^2-x+1=0$$ does not admit $$x=-2$$ as a triple root.
0
votes
0answers
31 views

Homomorphisms from $k[x,y]$ to $k[x,x^{-1},y]$

Let $k$ be a field of characteristic zero and let $f$ be a $k$-algebra homomorphism from $k[x,y]$ to $k[x,x^{-1},y]$. Denote $u:=f(x)$ and $v:=f(y)$. Assume that $u=x+\tilde{u}$ and $v=y+\tilde{v}$, ...
6
votes
5answers
2k views

Why is “division by $(z-1)$” valid here?

Is there an easy way to justify: $$x(x-1)(x+1) \equiv x(x^2-1) \Rightarrow (x-1)(x+1) \equiv x^2-1,$$ even for $x=0$? I seemingly have to divide by $x$ which should place the restriction $x \neq 0$ on ...
1
vote
0answers
49 views

Expressing a positive polynomial as the sum of two squares

I do not think this has been asked before, as I could not find anything when I answered this question (given for context, my question is self-contained). It is easy to show that a polynomial $p(x)\in ...
0
votes
0answers
12 views

Change of variables between roots and coefficients of monic polynomial

I have been reading Greg Anderson's "A Short Proof of Selberg's Integral Formula", though what I write here will be self-contained. I am stuck on a particular step. Suppose one has a monic poynomial $$...
1
vote
1answer
46 views

Find the remainder of a high degree polynomial

If $$f(x)=(x-1)^{2017}+(x-3)^{2016}+x^2+x+1$$ and $$g=x^2-4x+4$$ find the remainder of f divided by g. I only found that $$g=(x-2)^2$$ but I don't know how to go further. If I set $$x=2$$ then $$f(2)...
1
vote
2answers
45 views

General square form of Taylor expansion polynomial

Suppose we have a Taylor expansion for a function $f$ with respect to t up to $M$-th order. $$ \begin{equation} T_M = \sum^M_{k=0}\frac{1}{k!}f^k(x)\Delta t^k = f(x) + f'(x)\Delta t + \frac{1}{2}f''(...
2
votes
1answer
69 views

High degree polynomial with complex roots only (no real roots)

Suppose a given polynomial $p(x)=a_0+a_1x^1+...+a_{2n}x^{2n}$ for $n\in{N}$. How can I show that if all roots are complex? note that all coefficients are real. I actually need the solution for ...
4
votes
1answer
50 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
30 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 $(x-2)(x+1)$, ...
1
vote
2answers
28 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 $\mathbb{R}[x]$ ...
1
vote
2answers
129 views

Solve the following system of equations - (7).

Solve the following system of equations (over the reals). $$\large \left\{ \begin{aligned} (x + y)^2 &= xy + 3y - 1\\ x + y &= \frac{x^2 + y + 1}{x^2 + 1}\end{aligned} \right.$$ From the ...
4
votes
3answers
58 views

Solve for $x, y \in \mathbb R$: $x^2+y^2=2x^2y^2$ and $(x+y)(1+xy)=4x^2y^2$

Solve the following system of equations. $$\large \left\{ \begin{aligned} x^2 + y^2 &= 2x^2y^2\\ (x + y)(1 + xy) &= 4x^2y^2 \end{aligned} \right.$$ From the system of equations, we have that ...
2
votes
0answers
33 views

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

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
1answer
26 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
28 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 $y$ ...
-1
votes
1answer
43 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
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
3
votes
4answers
152 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
242 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
27 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 ...
5
votes
4answers
65 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
27 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
30 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 ...
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
40 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
125 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
votes
1answer
11 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
15 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
2answers
25 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
36 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
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?
-1
votes
0answers
35 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
52 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
38 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
45 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 ...
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 ...
2
votes
0answers
17 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
42 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
19 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
29 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
51 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 ...