Polynomial Definition and 1000 Threads

My solution: 1. ##K(a^2) \subseteq K(a)##. 2. ##a## is zero of the quadratic polynomial ##X^2 - (a^2)##, i.e., ##[K(a) : K(a^2)] \leq 2##. 3. It is not 2 because ##[K(a) : K] = [K(a) : K(a^2)][K(a^2) : K]## is odd. 4. Thus, it is 1, and hence ##K(a) = K(a^2)##. Is this solid enough?

This is a pattern I noticed when playing around with Mathematica. Is there any way to rigorously prove this? I was not able to find any literature concerning the number of factors in a finite field, especially because this is called a "pentanomial" in said literatures. These don't have much...

By IVT and trial and error, I get the interval to be ##(-\frac{1}{2},-\frac{1}{4})## I don't know how to do the next part. Let the actual root of the polynomial be ##x_{0}## and the approximate value is ##p##, we have ##|p-x_{0}|<\frac{1}{8}## I am not sure how to continue. Thanks

For this problem, The solution is, My solution is, Where I solved these equations to find where the function is ##f(x) > 0## and ##f(x) < 0##. Using ##x^2(1 - x) \geq 0## and ##x^2(1 - x) < 0## First equation: ##x^2 \geq 0 \implies x \geq 0## and ## 1 \geq x## Second equation: ##x^2(1...

How can you rewrite polynomial in terms of (x-a) instead of x? One thing came to mind is rewrite each x as x-a+2 (So it is x-2+2 in our example) but this will take long time and a lot of algebra steps, how did they do it very fast in the attached picture? thanks

Consider, I am self-studying; My interest is on the horizontal asymptote, now considering the degree of polynomial and leading coefficients, i have ##y=\dfrac{2}{1} =2## Therefore ##y=2## is the horizontal asymptote. The part that i do not seem to get is (i already checked this on desmos)...

What's the root formula for fifth and higher degree polynomial equations, which have roots in radicals?

When I look at the left hand side of the equation in above question then I can see that the highest degree of x would be 6 after the denominators are eliminated. I know that a polynomial of degree n will have n roots, but this one is not a pure polynomial since there is also a trigonometric...

Its a bit clear; i can follow just to pick another polynomial say ##(x+1)^3## are we then going to have ##(2x-2)+ x+3##? or it has to be a polynomial with ##x^2+1## being evident? cheers...

Hello! Let $n$ be a natural number, $P_n(x)$ be a polynomial with rational coefficients, and $\deg P_n(x) = n$. Let $P_0(x)$ be a constant polynomial that is not equal to zero. We define the sequence ${P_n(x)}_{n \geq 0}$ as an Appell sequence if the following relation holds: \begin{equation}...

My interest is on the highlighted; my understanding is that, let ##f(x)=x^3+x^2+2^{'}## then ##f(1^{'})=1{'}+1{'}+2^{'}=4^{'} ## we know that in ##\mathbb{z_3} ## that ##\dfrac{4}{3}=1^{'}## ##f(2^{'})=8^{'}+4^{'}+2^{'}=14{'} ## we know that in ##\dfrac{14^{'}}{3}=1^{'}##... I hope...

It is generally well-known that a plane algebraic curve is a curve in ##\mathcal{CP}^{2}## given by a homogeneous polynomial equation ##f(x,y)= \sum^{N}_{i+j=0}a_{i\,j}x^{i}y^{j}=0##, where ##i## and ##j## are nonnegative integers and not all coefficients ##a_{ij}## are zero~[1]. In addition, if...

Let ##Q_{n}(x)## be the inverse of an nth-degree polynomial. Precisely, $$Q_{n}(x)=\displaystyle\frac{1}{P_{n}(x)}$$, It is of my interest to use the set notation to formally define a number, ##J_{n}## that provides the maximum number of solutions of ##Q_{n}(x)^{-1}=0##. Despite not knowing...

Am going through this notes...kindly let me know if there is a mistake on highlighted part. I think it ought to be; ##α^2=5+2\sqrt{6}##

Hello, Am re-studying math & calculus aiming to start pure math studying later. However, I got this problem in Stewart calculus. Typically, this is a straightforward IVT application. x = x^3 + 1, call f(x)= x^3 - x + 1 & apply IVT. However I have two things to discuss. First thing is simple...

Let ##a,\,b,\,c## and ##d## be any four real numbers but not all equal to zero. Prove that the roots of the polynomial ##f(x)=x^6+ax^3+bx^2+cx+d## cannot all be real.

Is there a good rubric on how to choose the order of polynomial basis in an Finite element method, let's say generic FEM, and the order of the differential equation? For example, I have the following equation to be solved ## \frac{\partial }{\partial x} \left ( \epsilon \frac{\partial u_{x}...

My approach; Let ##(n+1)=k## ##(x-n+1)^3-(x-n)^3=3x^2+ax+b## ##(x-k)^3-(x-n)^3=3x^2+ax+b## ##(x^3-3x^2k+3xk^2-k^3)-(x^3-3x^2n+3xn^2-n^3)## ##=-3x^2k+3x^2n+3xk^2-3xn^2-k^3+n^3≡3x^2+ax+b## ##⇒1=n-k## ##⇒3(k^2-n^2)=a## ##⇒n^3-k^3=b## ##3(k+n)(k-n)=a## ##-3(k+n)=a## ##a=-3(n-1+n)## ##a=-3(2n-1)##...

I first checked for rational roots for this polynomial. The options are ##x=\pm 1/7##, both don't nullify the polynomial thus this polynomial doesn't have rational roots. Now, if it's reducible the only possible factors are: ##(ax^3+bx^2+cx+d)(Ax^3+Bx^2+Cx+D)=7x^6-35x^4+21x-1## or a product of...

Why is a third degree polynomial called a cubic polynomial? I just don’t see the connection between 3 and a cube.

Is this a polynomial? y = x^2 + sqrt(5)x + 1 I was told NO, the coefficients had to be rational numbers. I this true? It seem to me this is an OK polynomial. I can graph it and use the quad formula to find the roots? so why or why not?

The Homework Statement reads the question. We have $$ \langle f,g \rangle = \sum_{k=0}^{n} f\left(\frac{k}{n}\right) ~g\left( \frac{k}{n} \right) $$ If ##f(t) = t##, we have degree of ##f## is ##1##, so, should I take ##n = 1## in the above inner product formula and proceed as follows $$...

We have Galois extension ## K \subset L ## and element ##\alpha \in L## and define polynomial $$f = \prod_{\sigma \in Gal(L/K)} (x - \sigma(\alpha))$$ Now we want to show that ## f \in K[x] ## which is relatively easy to see because we can take ##\phi(f)## for any ## \phi \in Gal(L/K) ## then...

I'm stuck in a part of my problem where I need to find the roots of this function which represent turning points for a non-ideal spring.

Hey guys, Nice to be on here. I have been banging my brain for the last two weeks trying to come up with an algebraic solution to the following question - to no avail. Any input would be MUCH appreciated! The problem is somewhat long but can be summarized as follows: Begin with the following...

AIUI, an algebraic is defined as a number that can be the solution (root) of some integer polynomial, and is any number that can be constructed via any binary arithmetic operation or unary root operation with arguments that are themselves algebraic numbers. I have been able to prove this for...

I have a very urgent question, this is the problem. I have no idea how to solve this. I don't even know where to start. This is urgent, please at least tell me what is the name of this kind of problem so I can look it up. Specifically, # 4, 6 and 8. I can guess #4 by dividing both sides by y...

Theorem: Let ## f(x), g(x) \in \mathbb{F}[ x] ## by polynomials, s.t. the degree of ## g(x) ## is at least ## 1 ##. Then: there are polynomials ## q(x), r(x) \in \mathbb{F}[ x] ## s.t. 1. ## f(x)=q(x) \cdot g(x)+r(x) ## or 2. the degree of ## r(x) ## is less than the degree of ## g(x) ## Proof...

GRAPH WITH VALUES: Sorry I have a small dilema, I don't know if this is a exponential or polynomial function. I'd think its exponential but it doesn't have same change of factors.

So, I had a discussion with a friend of mine, neither of us are in controls but I was curious about an answer here. In a PID controller, we essentially take in an error value, do a mathematical operation on it and determine the input (controller output signal B) needed to the actuator to produce...

Hey! :giggle: I am looking at the following: a) Create a class QuadraticPolyonym that describes a polynomial of second degree, i.e. of the form $P(x)=ax^2+bx+c, a\neq 0$.The coefficients have to be givenas arguments at the construction ofan instance of the class. Implement a method...

This is exercise 12.1.2 a from Arfken's Mathematical Methods for Physicists 7th edition :Starting from the Laguerre ODE, $$xy''+(1-x)y'+\lambda y =0$$ obtain the Rodrigues formula for its polynomial solutions $$L_n (x)$$ According to Arfken (equation 12.9 ,chapter 12) the Rodrigues formula...

hey everyone . I want to plot a Grade 4 equation in MATLAB. but don't know how to do. Can anyone guide me? equation : f = 1.47*(x^4)-10^7*(x)+58.92*(10^6)

Hey! :giggle: Let $U\subset \mathbb{R}^n$ be an open set and $f:U\rightarrow \mathbb{R}$ is a $k$-times continusouly differentiable function. Let $x_0\in U$ be fixed. The $k$-th Taylor polynomial of $f$ in $x_0$ is $$T_k(x_1,\ldots ,x_n)=\sum_{m=0}^k\frac{1}{m!}\sum_{i_1=1}^n \ldots...

Hi, I was trying to find roots of the following cubic polynomial and there are only two roots. I believe there should be three roots. Could you please guide me why there are only two roots? If you say that the "1" repeats itself as a root then I'd say the same could be said of "0.9". Thank...

Hi, I did the first degree curve fitting in MATLAB. Please see below which also shows the output for each code line. But I wasn't able to get the same answer using Cramer's rule method presented below. I'm sure MATLAB answer is correct so where am I going wrong with the Cramer's rule method...

Let $p,\,q$ and $r$ be the distinct roots of the polynomial $x^3-22x^2+80x-67$. It is given that there exist real numbers $A,\,B$ and $C$ such that $\dfrac{1}{s^3-22s^2+80s-67}=\dfrac{A}{s-p}+\dfrac{B}{s-q}+\dfrac{C}{s-r}$ for all $s\not \in \{p,\,q,\,r\}$. What is...

Wish to determine when a system of polynomials has an infinite number of solutions, that is, is not zero-dimensional. The Wikipedia article : System of polynomial equations states: I interpret the quote to mean the system has an infinite number of solutions if the Grobner basis does not have...

The remainder of p(x)=x^3+ax^2+4bx-1 divided by x^2+1 is –5a + 4b. If the remainder of p(x) divided by x + 1 is –a – 2, the value of 8ab is ... A. -\frac34 B. -\frac12 C. 0 D. 1 E. 3 Dividing p(x) by x^2+1 by x^2+1 with –5a + 4b as the remainder using long division, I got (4bx – 1) – ((a – 1)x...

Suppose ##a_0+a_1x+\ldots+a_nx^n=0## and restrict the domain of ##p## to the set of real numbers excluding the roots of ##p##. Note that: if ##a_0 == 0##: ##x=0## is a root of ##p## else: ##x=0## is not a root of ##p## Assume the latter. Subtract ##a_0## from both sides of the equation...

Now $\sum_{i=0}^{10}(x_{i}+1) L _{10,i}(5) = (x_{0}+1) L _{10,0}(5) + (x_{1}+1) L _{10,1}(5) + ... + (x_{10}+1) L _{10,10}(5)$ Which I can further decompose into $\frac{(x_{0}+1)(5-x_{1})(5-x_{2})...(5-x_{10})}{(x_{0}-x_{1})(x_{0}-x_{2})...(x_{0}-x_{10})} +...

I consider three cases, based on the sign of ##a_0##. if ##a_0 == 0##: Set ##x=0##. \begin{align*} f(0)&=&a_0+a_1\cdot 0+a_2\cdot 0^2+a_3\cdot0^3+a_4\cdot0^4+0^5\\ &=&a_0+0\\ &=&0+0\\ &=&0 \end{align*} elif ##a_0<0##: Define ##M=\max\{|a_i|:1\leq a_i\leq 5\}## and set ##x=5(M+1)\neq 0##...

Consider an example polynomial: $$ \begin{align*} P_{16}(z)&=0.0687195 z^{16}+0.787411 z^{15}+4.58749 z^{14}+17.7271 z^{13}+50.5007 z^{12}\\ &+111.995 z^{11}+199.566 z^{10}+291.128 z^9+351.292 z^8+351.927 z^7+292.066 z^6\\ &+199.046 z^5+109.514 z^4+47.2156 z^3+15.1401 z^2+3.25759 z+0.362677...

Prove that the polynomial $P(x)=x^{13}+x^7-x-1$ has only one positive zero.

I say the answer is A.

Hello everyone. I have recently read the following article (which title is SAMBA: Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos) since I have some data in the form of a histogram without knowing the probability distribution function of said data. I have been able to calculate...

Let $a$ and $b$ be real numbers such that $a\ne 0$. Prove that not all the roots of $ax^4+bx^3+x^2+x+1=0$ can be real.

For ##x=-1## to be an *horizontal* inflection point, the first derivative ##y'## in ##-1## must be zero; and this gives the first condition: ##a=\frac{2}{3}b##. Now, I believe I should "use" the second derivative to obtain the second condition to solve the two-variables-system, but how? Since...

I was looking at this discussion of swapping roots of a polynomial causing the discriminant to loop around the origin. https://www.akalin.com/quintic-unsolvability Although it appears to be the case, has this mathematical fact ever been proven? It seems that the formula for the discriminant...