Polynomial Definition and 1000 Threads

i want to write a hypergeometric function (2F1(a,b;c,x)) as function of n that generate polynomials below n=0 → 1 n=1 → y n=2 → 4(ω+1)y^2-1 n=3 → y(2(2ω+3)y^2-3) n=4 → 8(ω+2)(2ω+3)y^4-6(6+4ω)y^2+3 ... → ... 2F1(a,b;c,x)=1+(ab)/(c)x+(a(a+1)b(b+1))/(c(c+1))x^2/2!+... the...

P(x) is polynomial of degree 2012, P(k)=2^k, k=0,1,...,2012. Find P(2013)

Like I mentioned in the title, this is probably one of the greatest challenge problems (I've seen so far) that designed for, hmm, well, for a challenge!:o Let $x_1$ be the largest solution to the equation $\dfrac{6}{x-6}+ \dfrac{8}{x-8}+\dfrac{20}{x-20}+\dfrac{22}{x-22}=x^2-14x-4$ Find the...

Proposition 5.2.1 in Artin states that: THEOREM. Let $p_k(t)\in \mathbf C[t]$ be a sequence of monic polynomials of degree $\leq n$, and let $p(t)\in \mathbf C[t]$ be another monic polynomial of degree $n$. Let $\alpha_{k,1},\ldots,\alpha_{k,n}$ and $\alpha_1,\ldots,\alpha_n$ be the roots...

Homework Statement If p(x) ∈F[x] is of degree 3, and p(x)=a0+a1∗x+a2∗x2+a3∗x3, show that p(x) is irreducible over F if there is no element r∈F such that a0+a1∗r+a2∗r2+a3∗r3 =0. Homework Equations The Attempt at a Solution Is this approach correct? If p(x) is reducible, then there...

1. What if absolute convergence test gives the result of 'inconclusive' for a given power series? We need to use other tests to check convergence/divergence of the powerr series but the matter is even if comparison or integral test confirms the convergence of the power series, we don't know...

I am reading Anthony W. Knapp's book, Basic Algebra. On page 146 in the section of Part IV (which is mainly on groups and group actions) which digresses onto rings and fields, we find the following text on the nature of ideals in the polynomial rings \mathbb{Q} [X] , \mathbb{R} [X] , \mathbb{C}...

A polynomial $P(x)=x^3+mx^2+nx+k$ is such that $n<0$ and $mn=9k$. Prove that the polynomial has three distinct real roots.

Let $p,\,q,\,r,\,s,\,t$ be distinct real numbers. Prove that the equation $(x-p)(x-q)(x-r)(x-s)+(x-p)(x-q)(x-r)(x-t)+(x-p)(x-q)(x-s)(x-t)+(x-p)(x-r)(x-s)(x-t)+(x-q)(x-r)(x-s)(x-t)=0$ has 4 distinct real solutions.

If $a,\,b$ are the two largest real roots of the polynomial $f(x)=3x^3-17x+5\sqrt{6}$, and their sum can be expressed as $\dfrac{\sqrt{m}+\sqrt{n}}{k}$ for positive integers $m,\,n,\,k$, find the value for $a+b$.

Let's say I wanted to do the following calculation: (x^2 + 2x + 1) / (x+1) I've scrolled through some online guides, and they all show how to do it, but not the principle behind it. I'm specifically having trouble with the fact, that instead of dividing the largest degree term with the entire...

Show that if $x^4+px^3+2x^2+qx+1$ has a real solution, then $p^2+q^2\ge 8$.

Does anyone know how to prove this identity? I don't quote understand why the associated Legendre function is allowed to have arguments where |x|>1. h_n(kr)P_n^m(\cos\theta)=\frac{(-i)^{n+1}}{\pi}\int_{-\infty}^\infty e^{ikzt}K_m(k\rho\gamma(t))P_n^m(t)\,dt where \gamma(t)=\begin{cases}...

Hi all, During my research i ran into the following general type of equation: \exp(ax+b)=\frac{cx+d}{ex+f} does anyone have an idea how to go about solving this equation? thx in advance

rewrite using polynomial long division \frac{x^3 + 4x^2 + 3}{x+4} so I did x+4 \sqrt{x^3 + 4x^2 + 0x + 3} and got x^2 + 1 - \frac{1}{x+4} What am i doing wrong? How is that incorrect?

Prove that $x^7-2x^5+10x^2-1$ has no root greater than 1. This is one of my all time favorite challenge problems! :o

Find all integral values of $k$ such that $q(a)=a^3+2a+k$ divides $p(a)=a^{12}-a^{11}+3a^{10}+11a^3-a^2+23a+30$.

Hi guys, My boss gave me a riddle. He says that you have a "black box" with a polynomial inside it like f(x)=a0+a1x+a2x^2+a3x^3 ... you don't know the rank of it or the coefficients a0, a1, a2 ... You do know: all of the coefficients are positive you get to input two x numbers and...

My question is hard of answer and the partial answer is in the wikipedia, but maybe someone known some article that already approach this topic and the answer is explicited. So, my question is: given: ##A = x_1 + x_2## ##B = x_1 x_2## reverse the relanship: ##x_1 = \frac{A +...

Homework Statement . Let ##X:=\{A \in \mathbb C^{n\times n} : rank(A)=1\}##. Determine a representative for each equivalence class, for the equivalence relation "similarity" in ##X##. The attempt at a solution. I am a pretty lost with this problem: I know that, thinking in terms of...

hi all I am stuyding how to factor equations such as :- 7x^2-9X-6 The problem i have is that it takes me too long to find 2 numbers whose sum is D and the same numbers whoes product is E. Is there any way/tips/guide on how i can achieve this quickly?

F = -0.2662*x^6 + 48.19*x^5 - 3424.2*x^4 + 121708*x^3 - 2*e^6*x^2 + 2*e^7*x - 6*e^7;

Question: What are the automorphisms of Z[x]? I know there are two automorphisms, one of which is the identity map, ø(f(x)) = f(x). What is the other one? ø(f(x)) = -f(x) for all f(x) in Z[x]? Or does it have something to do with the degree or factorization of the polynomials? Please...

If a polynomial of 1 variable, for example: P(x) = ax²+bx+c, can be written as P(x) = a(x-x1)(x-x2), so a polynomial of 2 variables like: Q(x,y) = ax²+bxy+cy²+dx+ey+f can be written of another form?

Hello! :) I am looking at the exercise: Prove that $f(x)=10x^4-18x^3+4x^2+7x+16 \in \mathbb{Z}[x]$ is irreducible in $\mathbb{Q}[x]$. According to my notes,a way to do this is the following: We know that $\forall m>1 \exists $ ring homomorphism $\widetilde{ \phi }: \mathbb{Z}[x] \to...

Problem: Show that the polynomial $x^8-x^7+x^2-x+15$ has no real root. Attempt: I am not sure what should be the best way to approach the problem. I thought of defining $f(x)=x^8-x^7+x^2-x$ because $f(x)+15$ is basically a shifted version of $f(x)$ along the y-axis. So if $15$ is greater than...

1. Is the root(det(q)) term in the Hamiltonian Constraint what makes it non polynomial 2. Is the motivation for Ashtekar Variables to remove the non polynomial terms by replacing the Hamiltonian with a densitised Hamiltonian

I'm not sure whether this should go in this forum or another. feel free to move it if needed Homework Statement Suppose that z_0 \in \mathbb{C}. A polynomial P(z) is said to be dvisible by z-z_0 if there is another polynomial Q(z) such that P(z)=(z-z_0)Q(z). Show that for...

Until now, I have avoided trying to display techniques of division using $\LaTeX$ because there just didn't seem to be a nice way to carry it out. However, we may use the array environment for the display of synthetic and polynomial long division methods very nicely. I will demonstrate how to...

If $P(x)=x^4+mx^3+nx^2+px+q$ is a polynomial whose roots are all negative integers, and given that $m+n+p+q=2009$, find $q$.

Homework Statement Let f(x), g(x), and h(x) be polynomials in x with real coefficients. Show that if (f(x))^2 −x(g(x))^2 =x(h(x))^2, then f(x) = g(x) = h(x) = 0. Find an example where this is not the case when we use polynomials with complex coefficients. i have no idea where to start...

Homework Statement problem in a pic attached Homework Equations The Attempt at a Solution i solved i and ii a , when it came to b , i just said that every one of the 3 roots will be squared having 2 roots 1 + and 1 - but then i read the marking schemes ( also attached) , and i got...

Homework Statement I am currently working through a chapter on Polynomial Remainder and Factor Theorems in my book, Singapore College Math, Syllabus C. There were a few problems which I got stuck on: 25) The positive or zero integer ##r## is the remainder when the positive integer...

Homework Statement Is f(x)=x^2+1 irreducible in \mathbb{F}_2[x] If not then factorise the polynomial. The Attempt at a Solution \mathbb{F}_2[x]=\{0,1\} f(0)=1 f(1)=1+1=0 Hence the polynomial is not irreducible

Homework Statement If we have the following data T = [296 301 306 309 320 333 341 349 353]; R = [143.1 116.3 98.5 88.9 62.5 43.7 35.1 29.2 27.2]; (where T = Temperature (K) and R = Reistance (Ω) and each temperature value corresponds to the resistor value at the same position) Homework...

Homework Statement Prove that ##(a-b)## is a factor of ##a^5-b^5##, and find the other factor. Homework Equations Remainder theorem : remainder polynomial ##p(x)## divided by ##(x-a)## is equal to ##p(a)## Factor theorem : if remainder = 0, then divisor was a factor of dividend...

Could someone help me with this question? Because I'm stuck and have no idea how to solve it & it's due tomorrow :( Let S be the following subset of the vector space P_3 of all real polynomials p of degree at most 3: S={p∈ P_3 p(1)=0, p' (1)=0} where p' is the derivative of p...

Homework Statement This is part of a linear algebra problem. I've found the characteristic polynomial of a matrix however it's a degree 4 polynomial and I'm having trouble solving it Homework Equations λ4+λ3-3λ2-λ+2 = 0 The Attempt at a Solution I replaced λ2 with a and did...

I am working on a circuit that inputs a 31-bit pseudo-random binary string into a CCIT CRC-16 block which generates a 16-bit CRC output. I know that M(x)/G(x) = Q(x) + R(x) and the transmitted code will be R(x) appended to M(x). When I simulated the circuit, I got a CRC of 1 0 1 0 1 0 0 1...

f(x)= x (x+2)^2 (x-1)^4Zeros would be: 0, -2, 1 Multipicity of : 1 2 4Then for y- intercept: f(0)=0 And don't know how to graph it...

Let $f(x)=x^4+px^3+qx^2+rx+s$, where $p,\,q,\,r,\,s$ are real constants. Suppose $f(3)=2481$, $f(2)=1654$, $f(1)=827$. Determine the value of $\dfrac{f(-5)+f(9)}{4}$.

Hey guys I am having a little bit of trouble with using and understanding the linear factorization theorem to find the polynomial function. Homework Statement Find an nth degree polynomial function with real coefficents satisfying the given conditions. n=3; -5 and 4+3i are zeros...

Two days ago, I was absolutely certain to have proved the theorem hereafter. But then, micromass pointed out that another theorem the truth of which I was also certain was in fact false. It seems that in mathematics, never be certain until other mathematicians are. This is the reason why I...

Show that the equation $8x^4-16x^3+16x^2-8x+p=0$ has at least one non-real root for every real number $p$ and find the sum of all the non-real roots of the equation.

Please see attached. From part (a), we know that kernel (D) is a constant function, i.e f(x)=c, say From part (d), we know that eigenvalue of D is zero My question: For part (e), Is it correct to say that D is diagonalizable if and only if P_n (R) = ker (D) ?? so the only solution is...

I do not understand how I would do this with long division since there is only 2 terms. I can't remember the trick. Here is what I have so far. \int \frac{3x^2 - 2}{x^2 - 2x - 8} dx so I got \int 3 + \frac{x^2 - 2}{(x - 4)(x + 2)} I'm not sure if that's right? I just factored it out instead...

Homework Statement A polynomial of degree two or less can be written on the form p(x) = a0 + a1x + a2x2. In standard basis {1, x, x2} the coordinates becomes p(x) = a0 + a1x + a2x2 equivalent to ##[p(x)]_s=\begin{pmatrix}a0\\ a1\\ a2 \end{pmatrix}##. Part a) If we replace x with...

Homework Statement Let f(x) = anxn + an-1xn-1 + ... + a1x + a0 be a polynomial where the coefficients an, an-1, ... , a1, a0 are integers. Suppose a0 is a positive power of a prime number p. Show that if \alpha is an integer for which f( \alpha ) = 0, \alpha is also a power of p. Homework...

Hello everyone! I have this polynomial: $p(x) =$ 1 + \sum_{k=1}^{13}\frac{(-1)^k}{k^2}x^k - I'm supposed to show that this polynomial must have at least one positive real root. - I'm supposed to show that this polynomial has no negative real roots. - And I'm supposed to show that if $z$ is...

Hi! I have also an other question (Blush) Knowing that $f$ is a polynomial function,how can I show that there is a $y \in \mathbb{R}$,such that $|f(y)|\leq |f(x)| \forall x \in \mathbb{R}$ ?