Polynomial Definition and 1000 Threads

##1+x+x^2 = \dfrac{1-x^3}{1-x} = (1-x^3)\cdot \dfrac{1}{1-x} = (1-x^3)\sum_{k=0}^\infty x^k##. Isn't this a contradiction since the LHS has degree ##2## while the RHS has infinite degree?

Hey! :o I want to determine an approximation of a cubic polynomial that has at the points $$x_0=-2, \ x_1=-1, \ x_2=0 , \ x_3=3, \ x_4=3.5$$ the values $$y_0=-33, \ y_1=-20, \ y_2=-20.1, \ y_3=-4.3 , \ y_4=32.5$$ using the least squares method. So we are looking for a cubic polynomial $p(x)$...

Homework Statement We were given a tutorial to complete which I did complete. Now the question is: By modifying the appropriate lines in your script file, find the values of a, b, c, and d so that the cubic polynomial  y = ax3 + bx2 + cx + d  passes through the (x, y) pairs (-1, 3), (0, 8)...

Homework Statement Given a polynomial p, let A be the sum of the coefficients of the even powers, and let B be the sum of the coefficients of the odd powers. Prove that A^2 - B^2 = p(1)p(-1). Homework EquationsThe Attempt at a Solution See attached. Can someone please look at my work to see if...

I would like to have someone who would be willing to explain me what is a like term and an unlike term in terms of set theory. I'm just an high-scool student, but I really would like to understand it from that point of view anyway. It doesn't have to be a 1000 pages long of explanations.

Hello everyone, Going through calculus study, there is a vague point regarding polynomials I'd like to make clear. Say there's a polynomial ##f## with a root at ##a## with multiplicity ##2##, i.e. ##f(x)=(x-a)^2g(x)## where ##g## is some other polynomial. I define ##h(x)=\frac {f(x)} {x-a}##...

Homework Statement ##3f(x)+2f(\frac{1}{x}) = x##, solve ##f(x)## Homework Equations Not sure.Maybe the ones of inverse functions. The Attempt at a Solution The only thing that I came up so far is that the function’s highest order term is ##x## because if there are higher orders,it will show...

Recently I came up with a proof of “ for a nth degree polynomial, there will be n roots” Since the derivative of a point will only be 0 on the vertex of that function,and a nth degree function, suppose ##f(x)##has n-1 vertexes, ##f’(x)## must have n-1 roots. Is the proof valid?

<Moderator's note: Moved from a technical forum and thus no template.> Task: http://snk066.tk/math/Task.png My solution: http://snk066.tk/math/my_solution.jpg What you need to? I need an answer in the form: u (x,t) = (some polynomial) The solution is not really necessary, if someone will...

$$\int x^2+3 = \frac{x^3}{3}+3x+C$$ I can get the front two part by power rule, but what is the C doing there? Wolframalpha suggested it should be a constant, but what value should it be? Sorry for asking rookie questions:-p

Homework Statement Define {x \choose n}=\frac{x(x-1)(x-2)...(x-n+1)}{n!} for positive integer n. For what values of positive integers n and m is g(x)={{{x+1} \choose n} \choose {m}}-{{{x} \choose n} \choose {m}} a factor of f(x)={{{x+1} \choose n} \choose {m}}? Homework Equations The idea...

Homework Statement Let Tn(x)=1+2x+3x^2+...+nx^(n-1) Find the value of the limit lim n->infinity Tn(1/8).The Attempt at a Solution How do I solve this? I know how to write the polynomial as a series, but not sure how if this is the best way of finding the limit.

Homework Statement The equation 4x = (1/3)*cos(3x) has a solution on the interval [0,1]. Find an approximative solution by replacing the right hand side with a Taylor polynomial of degree 2 around 0. Homework EquationsThe Attempt at a Solution So as I understand the task we should find a...

Hey! :o Let $K$ be a normal extension of $F$ and $f\in F[x]$ be irreducible over $F$. Let $g_1, g_2$ be irreducible factors of $f$ in the ring $K[x]$. Show that there exists $\sigma \in G(K/F)$ such that $g_2=\sigma (g_1)$. If $f$ is reducible over $K$, show that all its irreducible...

Homework Statement -x^3 - 6x^2 -12x -8 Homework EquationsThe Attempt at a Solution I don't know, I just know the roots are -2 with multiplicity 3.

Homework Statement Find all ##a,b,c\in\mathbb{R}## for which the zeros of the polynomial ##az^3+z^2+bz+c=0## are in this relation $$z_1^3+z_2^3+z_3^3=3z_1z_2z_3$$ Homework Equations we know that if we have a polynomial of degree 3 the zeroes have relation in this case ##z_1+z_2+z_3=-1/a##...

Homework Statement Determine whether there exist ##A## and ##B## such that: $$\frac{1}{3x^2-5x-2} = \frac{A}{3x+1} + \frac{B}{x-2}$$Homework Equations None The Attempt at a Solution [/B] First I divided the polynomial ##3x^2-5x-2## by ##3x+1## and got ##x-2## as a result without a...

First time in this forum, so greetings to everyone! I am currently working with some physical models in the field of natural ventilation and I came across the following 5th order polynomial equation (quintic function): $X^{5}+ C X - C =0$ This is the steady state solution of a physical system...

Homework Statement >Find the sum of the roots, real and non-real, of the equation x^{2001}+\left(\frac 12-x\right)^{2001}=0, given that there are no multiple roots. While trying to solve the above problem (AIME 2001, Problem 3), I came across three solutions on...

Homework Statement "Show that for some ##x\in ℝ##, that ##x^5+2x^4+3x^3+2x^2+x=1##." Homework EquationsThe Attempt at a Solution Okay, so I know from Descartes' rule of sign that the function ##f(x)=x^5+2x^4+3x^3+2x^2+x-1## has exactly one positive root, since the sign of the coefficients...

Hey! :o I want to show that the polynomial $x^4-2\in \mathbb{Q}[x]$ remains irreducible in the ring $\mathbb{Q}(i)[x]$. I have done the following: The polynomial is irreducible in $\mathbb{Q}[x]$ by Eisenstein's criterion with $p=2$. Then if $a$ is a root of $x^4-2$ then the degree of the...

Question $$\int_{-1}^{1} cos(x) P_{n}(x)\,dx$$ ____________________________________________________________________________________________ my think (maybe incorrect) $$\int_{-1}^{1} cos(x) P_{n}(x)\,dx$$ $$\frac{1}{2^nn!}\int_{-1}^{1} cos(x) \frac{d^n}{dx^n}(x^2-1)^n\,dx$$ This is rodrigues...

This time my struggle is with ring ideals. Book still won't provide examples, so I'm again trying to come up with some of my own. I figured {0,2} might fit the definition as an ideal of ##\mathbb{Z/4Z}## since it is an additive subgroup and ##\forall x \in I, \forall r \in R: x\cdot r, r\cdot x...

Hello! (Wave) Let $\mathbb{R}[x]_{ \leq n}$ be the vector space of the real polynomials of degree $\leq n$, where $n$ a natural number. I want to show that there is a unique $q(x) \in \mathbb{R}[x]_{\leq n}$, with the property that $\int_{-1}^1 p(x) e^x dx=\int_0^1 p(x) q(x) dx$, for each $p(x)...

Hello! (Wave) If the matrix $A \in M_n(\mathbb{C})$ has $m_A(x)=(x^2+1)(x^2-1)$ as its minimal polynomial, then I want to find the minimal polynomials of the matrices $A^2$ and $A^3$. ($M_n(k)$=the $n \times n$ matrices with elements over the field $k=\mathbb{R}$ or $k=\mathbb{C}$) Is there a...

11. Given a polynomial with the degree 3. If it is divided by x^2+2x-3, the remainder is 2x + 1. If it is divided by x^2+2x, the remainder is 3x - 2. The polynomial is ... A. \frac23x^3+\frac43x^2+3x-2 B. \frac23x^3+\frac43x^2+3x+2 C. \frac23x^3+\frac43x^2-3x+2 D. x^3+2x^2+3x-2 E. 2x^3+4x^2+3x+2...

I've been trying to prove the impossibility of the quintic "on the cheap" without having to go through a graduate course in abstract algebra (I haven't even done the undergraduate course, although I've been reading up on it a little bit at a time). I understand Bezout's Lemma, with a practical...

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Hello. This is the differential equation. $$ i \cdot \dfrac{R}{L} + \dfrac{di}{dt} = t $$ My solution path: Homogenous solution: $$ r + \dfrac{R}{L} = 0 \\ \\ r = -\dfrac{R}{L} \\ \\ i_{h}(t) = C_{0} e^{ -\dfrac{R}{L} t} \\ \\ $$ Particular solution: Try $$ y_{p} = at + b \\ \\ y_{p}...

Homework Statement Looking to factor ##-2x^3-3## and having an issue. To my understanding, the Fundamental Theorem of Algebra tells us that it is at least theoretically possible to factor any polynomial of degree n. Homework EquationsThe Attempt at a Solution So my first step to factor this...

Hello, I have been going through the Wisconsin Placement Exam sample test. I'm trying to figure out how to find the solution set for x6 – 7x3 + 12. I have tried having u = x3 and solving for u2-7u+12, but I'm unsure what to do once I get (u - 4)(u - 3). Would someone help me figure out how to...

I have the integral ##\int_{-\infty}^{\infty} x^2 e^{-x^2} ~dx##. Is there any simple way to integrate this, given that that I already know that the value of the Gaussian integral is ##\sqrt{\pi}##?

Why is it that for a 7th degree polynomial, the number of real roots is either 1, 3, 5, or 7?

Homework Statement Factor ##x^4-3x^2+9## over the reals Homework EquationsThe Attempt at a Solution I am factoring this polynomial over the reals. So there are two options. It will either split into two linear factors and an irreducible quadratic, or two irreducible quadratics. I'm really not...

I have the simple quartic polynomial ##x^4+1##. How in general do I determine whether this is factorable over the reals or not? Since it has no real roots, it could only factor into two quadratic polynomials, but I am not sure what I can do to determine whether this is possible or not.

Hey, first off, I'm not sure if this is the right section. If another section is better, please let me know and I'll be more careful next time. So, my problem is with a degree 3 complex polynomial. I'm given one zero of the equation, but since it is a complex zero, I can use the conjugate too...

Homework Statement Find roots of $$ -\lambda ^3 +(2+2i)\lambda^2-3i\lambda-(1-i) = 0 $$ Homework EquationsThe Attempt at a Solution I tried my old trick I tried to separating the 4 terms into 2 pairs and try to find a common factor in the form of ##\lambda + z## between them, $$ -\lambda ^2...

The graph below shows a portion of the curve defined by the quartic polynomial P(x) = x^4 + ax^3 + bx^2 + cx + d. Which of the following is the smallest?https://imgur.com/a/1VuGSiA(A) P(-1) (B) The product of the zeros of P (C) The product of the non-real zeros of P (D) The sum of the...

Homework Statement Show that ##\displaystyle \sum_{i=0}^{100} {100\choose i}{200-i\choose 198-i}x^i## is divisible ##(x+1)^{98}##. Homework EquationsThe Attempt at a Solution I am pretty stumped, but I have a few general. I think that the the binomial theorem will be involved. That is, I think...

I am really struggling on the following Algebra question: Consider the Irreducible Polynomial g = X^4 + X + 1 over 𝔽2 and let E be the extension of 𝔽2 = {0,1} with root α of g. (a) How many elements does E have? (b) Is every non-zero element of E of the form α^n with n ϵ N (natural numbers)...

Homework Statement Show that A is a scalar matrix kI if and only if the minimum polynomial of A is m(t) = t-k Homework EquationsThe Attempt at a Solution f(A) is monic f(A) = 0 since A = kI Next we must show that deg(g) < deg(f) I guess I'm not sure where g comes from. Is it merely an...

Homework Statement Express a polynomial in terms of the basis vectors. {x2 + x, x + 1, 2} Homework Equations 3. The Attempt at a Solution [/B] I think the answer is: (x2+x)^2 + (x + 1) + 2 = 0 simplified to become: x4 + 2x3 + x2 + x + 3 = 0

I would like to have verification if the following attached proof is correct. If it is not correct, what can be done to make it correct? Thanks.

Given a prime number $p$, prove that the polynomial congruence $(x + y)^n \equiv x^n + y^n$ (mod $p$) is true if and only if $n$ is a power of $p$.

Please bare with me. Most of you know I actually don't have a great math background. In any case I'm going way back and filling in some very basic math that I have long forgot. I have some questions about terms in a polynomial. Here is an example $$3x^5+7x^3-5$$ 1. From my book 3 and 7 are...

Hi everybody. In Python there is a library called chaospy. One useful command is cp.orth_ttr which generates a polynomial expansion, e. g. a series of orthogonal polynomials or orders zero, one, two... for a random variable e.g normal, uniform... For more information see...

Just a general question here. So for a polynomial function, the behavior of the graph at the zeros is determined by the evenness or oddness of the magnitude of the zeros. If the magnitude is odd, the graph will cross the zero. If the magnitude is even, it will bounce at the zero. Why is this...

A polynomial f(x) = 2x^3-5x^2+ax+18 is divisible by (x - 3). The result of that polynomial f(x) divided by (x - 1) is ... A. 2x^2-7x+2 B. 2x^2+7x-2 C. 2x^2-7x-2 D. x^2-6x-2 E. x^2-6x+3 I got a + 3 = -6 and so a = -9 and f(x) = 2x^3-5x^2-9x+18, but when I divided it with x - 1 I got x^2-3x+6...

One of the solutions to x4-2x3+kx2+px+36 = 0 is x = 3[FONT=times new roman]i [FONT=arial]Prove that this polynomial has no real solutions (roots) and find the real values of k and p...

Homework Statement ## x^2 +xy + y^2 = 3## ## y^2+yz+z^2=1## ##z^2+zx+x^2=4##Homework EquationsThe Attempt at a Solution ##yz^2-xz^2+yx^2-xy^2=4y-x## ##z^2(y-x)-xy(y-x)=4(y-x)## thus ##z^2-xy=4## or ##z^2=4+xy## ......on working out i end up with ## z^4-8z+16+(z^2-4)y^2+y^4= 3y^2## and ##...