Polynomial Definition and 1000 Threads

Hi, Homework Statement I am expected to show that the polynomial a1xb1 + a2xb2 + ... + anxbn = 0 has at most n-1 solutions in (0,infinity), where a1,a2,...,an are real numbers different than zero, and b1,b2,...,bn are real numbers so that bj is different than bk for j different than k...

Homework Statement http://puu.sh/1QFsA Homework Equations The Attempt at a Solution I'm actually not sure how to do this question. How do i find Δx^2. I kind of understand the question but I don't know how to prove it. I know that Δy becomes dy when the width becomes...

Hello all, I have a series of polynomials P(n), given by the recursive formula P(n)=xP(n-1)-P(n-2) with initial values P(0)=1 and P(1)=x. P(2)=xx-1=x2-1 P(3)=x(x2-1)-(x)=x3-2x ... I am confident that all of the roots of P(n) lie on the real line. So for P(n), I hope to find these roots. I...

Homework Statement Find all polynomials of the form a + bx + cx^2 that: Goes through the points (1,1) and (3,3) and such that f'(2) = 1Homework Equations a + bx + cx^2 f'(x) = x+2cx f'(2) = 2 + 4c polynomial through (1,1) = a + b1 + c1 = 1 polynomial through (3,3) = a + b3+ c3^2 = 3 The...

Hi, Homework Statement I am asked to prove that given all roots of a polynomial P of order n>=2 are real, then all the roots of its derivative P' are necessarily real too. I am permitted to assume that a polynomial of order n cannot have more than n real roots. Homework Equations...

Homework Statement Last exam in my school this exircise was given: From norweagen: " Decide the Taylor polynomial of second degree of x=0 of the function: f(x) = 3x^3 + 2x^2 + x + 1 I found the Taylor polynomial of second degree to be: 2X^2+X+1, which is correct. If I get an...

Hello everyone, first time poster, long time reader here! I'm an ex-math major and while I'm no longer pursuing a degree anymore in mathematics, I still continue onwards in my spare time trying to learn as much as I can about it because it's always been something I've enjoyed partaking in and...

Hi Let p(x,y)≥0 be a polynomial of degree n such that p(x,y)=0 only for x=y=0.Does there exist a positive constant C such that the inequality p(x,y)≥C (IxI+IyI)^n (strong inequality!) holds for all -1≤x,y≤1? The simbol I I stands for absolute value.

Homework Statement Given some ElGamal private key, and an encrypted message, decrypt it. Homework Equations Public key (F_q, g, b) Private key a such that b=g^a Message m encrypted so that r=g^k, t=mb^k Decrypt: tr^-a = m The Attempt at a Solution My problem is finding r^-a...

Ok I promise this time it is not a homework type question. If someone could direct with a name of a theorem here, then I'll go ahead and google it, otherwise have a look. I have a chunk of notes that I'm confused about. We're not shown any proofs here or any explanations, just what is seen. It...

Here is a very difficult cubic polynomial. x^3 - x - 2 = 0 I am wondering whether it is solvable or not. Please think about it.

Homework Statement Given that the minimal polynomial of a over rationals is x^4+x+8, find the minimal polynomial for 1/a over Q. Homework Equations I know there is a lot of work done out there for finding the min. polynomials of a^k for k>0, however I've never seen anything with a^k for...

Hey, guys. Having problems with this question because I don't exactly know how to begin it. Homework Statement The problem states to: "Find the Taylor polynomial of smallest degree of an appropriate function about a suitable point to approximate the given number to within the indicated...

Homework Statement I'm having a bit of trouble with this Maclaurin Series question. It should be simple enough but I can't get the answer which is given as x2. It's been a while since I've done series and my being rusty is a little annoying. Hopefully someone can help :) Consider...

For the years 1998-2009, the number of applicants to US medical schools can be closely approximated by: A(t)= -6.7615t4+114.87t3-240.1t3-2129t2+40,966 where t is the number of years since 1998. a) graph the number of applicants on 0<= t <= 11 b) based on the graph in part a, during what...

Hi. There is a polynomial f = (x^3) + 2(x^2) + 1, f belongs to Q[x]. It will be shown that the polynomial is irreducible by contradiction. If it is reducible, (degree here is three) it must have a root in Q, of the form r/s where (r,s) = 1. Plugging in r/s for variable x will resolve to r^3 +...

Hi. There is a polynomial f = (x^3) + 2(x^2) + 1, f belongs to Q[x]. It will be shown that the polynomial is irreducible by contradiction. If it is reducible, (degree here is three) it must have a root in Q, of the form r/s where (r,s) = 1. Plugging in r/s for variable x will resolve to r^3 +...

-x^3+12x+16 Every single technique I read about online of how to factor 3rd degree polynomials, it says to group them. I don't think grouping works with this. I tried but it didn't work, since there's only 3 terms. Apparently I'm not supposed to have a cubic variable without a squared...

Let k[x,y,z,t] be the polynomial ring in four variables and let I=<x,y>, J=<z, x-t> be ideals of the ring. I want to show that IJ=I \cap J and one direction is trivial. But proving I \cap J \subset IJ has stumped me so far. Anyone have any ideas?

Homework Statement Show that the descriminant of the characteristic polynomial of K is greater than 0. K=\begin{pmatrix}-k_{01}-k_{21} & k_{12}\\ k_{21} & -k_{12} \end{pmatrix} And k_i > 0 Homework Equations b^2-4ac>0 The Attempt at a Solution I have tried the following...

Homework Statement Given the function "P" defined by: P(x) := [SIZE="3"]x^2n + [SIZE="3"]a2n-1*[SIZE="3"]x^2n-1 + ... + [SIZE="3"]a1[SIZE="3"]x + [SIZE="3"]a0; prove that there exists an [SIZE="3"]x* in |R such that P(x*) = inf{P(x) : x belongs to | R} Also, prove that: |P(x*)| =...

I am not going to post my question because I want to find out how to actually use the taylor polynomial and morse potential and then apply that to my problem. Say I have to approximate the morse potential using a taylor series expanding about some value. This will then find me the force...

I've been given a matrix A and calculated the characteristic polynomial. Which is (1-λ)5. Given this how does one calculate the minimal polynomial? Also just to check, is it correct that the minimal polynomial is the monic polynomial with lowest degree that satisfies M(A)=0 and that all the...

Homework Statement Knowing that the equation: X^n-px^2=q^m has three positive real roots a, b and c. Then what is log_q[abc(a^2+b^2+c^2)^{a+b+c}] equal to? Homework Equations a + b + c = -(coefficient \ of \ second \ highest \ degree \ term) = -k_2 abc = -(constant \ coefficient) =...

$\beta m^5 + m^2 + 1 =0$ How do I find the roots?

Here is the correct answer: 2x^3 - x^2 + 4x - 5 My attempt only gives me one cubed term and the other terms are also marginally off, any help on who can show me how to get the correct answer will be hugely appreciated

I am reading Jackson's electrodynamics book. When I went through the Legendre polynomial, I have a question. In the book, it stated that from the Rodrigues' formula we have Consider only the odd terms...

Homework Statement z^4 - z^2 + 1 = 0 is an equation in ℂ. Which of the following alternatives is the sum of two roots of this equation: (i) 2√3; (ii) -(√3)/2; (iii) (√3)/2; (iv) -i; (v) i/2 Homework EquationsThe Attempt at a Solution All I know is that the sum of all roots should equal 0...

Homework Statement The condition that x^4+ax^3+bx^2+cx+d is a perfect square, is Homework Equations The Attempt at a Solution If the above polynomial will be a perfect square then it can be represented as (x-\alpha)^2(x-\beta)^2 where α and β are the roots of it.This means that two...

Find the sequence (B_nf) of Bernstein's polynomials in a) f(x)=x and b) f(x)=x^2 Answers (from my textbook): a) B_nf(x) = x for all n. b) B_nf(x) = x^2 + \frac{1}{n} x (1-x) I know that the bernstein's polynomial is: B_nf(x) = \sum_{k=0}^n f (\frac{k}{n}) \binom{n}{k} x^k...

Homework Statement Find the third degree Taylor polynomial about the origin of f(x,y) = \frac{\cos(x)}{1+xy} Homework Equations The Attempt at a Solution From my ventures on the Internet, this is my attempt: I see that \cos(x) = 1 - \frac{1}{2}x^2 + \frac{1}{4!}x^4 - \cdots...

Is there a formula for finding the roots of a bivariate polynomial in x and y with the form: (a^2)xy+abx+acy+bc Where a, b, and c are constants, of course.

hey i'm trying to figure out how to approach part b of this problem, http://imageshack.us/a/img850/6059/asdasdno.jpg so i can see that you can apply the mean value theorem to p'(x) so there exists some c between a and b such that f'(c) = [f(b) - f(a)] / (b-a)=0 so p'(x)...

is there any general formula to find out zeros of a cubic polynomial that will give you all the zeros ? if not please tell me what are the different methods to find out the zeros , guessing and trial and error , numerical etc. i want to see where are each methods useful and is there...

Question: 1.Find the range of values of \(a\) for which \[(2-3a)x^2+(4-a)x+2=0\]has real roots.2. If the roots of the equation \(4x^3+7x^2-5x-1=0\) are \(\alpha\) , \(\beta\) and \( \gamma\),find the equation whose roots are: (a) \( \alpha+1,\beta+1\) and \(\gamma+1\) (b) \(\alpha^2 \beta^2\)...

Find a polynomial p(t) of degree 6 which has a zero of multiplicity 2 at t = 1 and a zero of multiplicity 3 at t = 2, and also satisfying: p(0) = 2 and p`(0) = 1. What is the other root of p(t)? Attempt at solution: zero of multiplicity 2 at t =1 implies (t-1)^2 is a factor or p(1) = 0...

Homework Statement Let's say that we have a second order polynomial function, and we know all of the points where it intersects with the x and y axis. Ex: (-2; 0), (0; 2), (1; 0) How does on determine the ax^2+bx+c polynomial form based on that? Homework Equations - The Attempt at...

Homework Statement The problems states "All polynomials of the form p(t)= at^2, where a is in R." I'm supposed to see if it is a subspace of Pn. I've already done that but the book's answer is that it spans Pn by Theorem 1, because the set is span{t^2} Homework Equations Theorem states "1 If...

Homework Statement Let f(x) = x^{4}+ax^{3}+bx^{2}+cx+d be a polynomial with real coefficients and real zeroes. If |f(i)| = 1, (where i = \sqrt{-1}) then find a+b+c+d. Homework Equations The Attempt at a Solution f(i) = 1-b+d+ci-ai Taking modulus |f(i)|= |1-b+d+i(c-a)|...

Homework Statement If we have a transformation matrix \begin{bmatrix} 1 & 2 & 4 \\0 & 0 & 0 \\0 & 0 & 0 \end{bmatrix} Homework Equations The Attempt at a Solution I found the characteristic polynomial of this matrix: x^3 - x^2 = x^2(x-1) ...can anybody please help me...

Homework Statement find the polynomial function p(x) with zeros, -1, 1, 3 and P(0)=9 Homework Equations all i have is (x^2-1) and (x-3) The Attempt at a Solution

Homework Statement if 2 is a root of 4x^3-3x^2-kx-4k^2= 0 find th value of k Homework Equations The Attempt at a Solution

The first thing that we should notice is that the leading coefficient $a_n = 1$. I was thinking about considering the factored form of p. I googled, and there is an algorithm called the "Schur-Cohn Algorithm" that is suppose to answer exactly this, but I can't find any information on it or...

Homework Statement Consider the vector space F(R) = {f | f : R → R}, with the standard operations. Recall that the zero of F(R) is the function that has the value 0 for all x ∈ R: Let U = {f ∈ F(R) | f(1) = f(−1)} be the subspace of functions which have the same value at x = −1 and x = 1...

Homework Statement Consider the vector space F(R) = {f | f : R → R}, with the standard operations. Recall that the zero of F(R) is the function that has the value 0 for all x ∈ R: Let U = {f ∈ F(R) | f(1) = f(−1)} be the subspace of functions which have the same value at x = −1 and x = 1...

Taking the derivative of a polynomial fraction?? b]1. Homework Statement [/b] Ok, so the question wants me to differentiate f(x)= (x)/(x+1). We are supposed to use the definition of the derivative f'(x)= (limit as h->0) [f(x+h)-f(x)]/(h). We also have learned the power rule. I did the formula...

Could someone quickly go over my working, as I am not 100% sure I have done it the right way. I will show and explain my working step by step. $$ at^2-4a + 2t^2-8$$ I first grouped the values: (at^2-4a) + (2t^2-8) I then factorised these equations into: a(t^2-4a) + 2(t^2-4) I...

Hello, This was an exam question which I wasn't sure how to solve: Suppose f is entire and |f(z)| \leq C(1+ |z|)^n for all z \in \mathbb{C} and for some n \in \mathbb{N}. Prove that f is a polynomial of degree less than or equal to n. I know that f can be expressed as a power series, but I'm...

Suppose there is a set of complex variables \{x_i,\,i=1 \ldots M;\;\;y_k,\,k=1 \ldots N\} and a polynomial equation p(x_i, y_k) = 0 Is there a way to prove or disprove for such an equation whether it can be reformulated as f(x_i) = g(y_k) with two functions f and g with...

Homework Statement Determine whether the following are subspaces of P4: a) The set of polynomials in P4 of even degree b) The set of all polynomials of degree 3 c) The set of all polynomials p(x) in P4 such that p(0) = 0 d) The set of all polynomials in P4 having at least one real root The...