Polynomials Definition and 740 Threads

Alright, I'll be honest. I was extremely tired and slept all through the lesson in Algebra today lol. And now I need help with factoring polynomials. Example problems that I need help on: 7h3+448 Perfect square factoring - y4-81 Grouping - 3n3-10n2-48n+160 You don't have to answer...

Homework Statement find the rank and nullity of the linear transformation T:U -> V and find the basis of the kernel and the image of T Homework Equations U=R[x]<=5 V=R[x]<=5 (polynomials of degree at most 5 over R), T(f)=f'''' (4th derivative) The Attempt at a Solution Rank = 2...

Homework Statement Suppose A is a 2x2 real matrix with characteristic polynomial f(t) = t2 - 5t +4. Find a real polynomial g(t) of degree 1 such that (g(A))2 = A. Suppose A is a 2x2 complex matrix with A2 ≠ O. Show that there is a complex polynomial g(t) of degree 1 such that (g(A))2 = A...

I have problem understand in one step of deriving the Legendre polymonial formula. We start with: P_n (x)=\frac{1}{2^n } \sum ^M_{m=0} (-1)^m \frac{2n-2m)}{m!(n-m)(n-2m)}x^n-2m Where M=n/2 for n=even and M=(n-1)/2 for n=odd. For 0<=m<=M \Rightarrow \frac{d^n}{dx^n}x^2n-2m =...

Nevermind Solved it.

Hello, First of all, I am not trying to "spam" subforums. I found out that my thread shouldn't be posted under homework. Anyways, here it is. Integration Let say there's a polynomial, 5x+6 and you want to integrate from 0 to 3 respect to x, how do you input in MATLAB? (I guess you can't...

Homework Statement Prove for each square matrix B there is a real polynomial p(x) (not the zero polynomial) so p(B)=0 Homework Equations Rank-nullity? dimv = r(T) + n(T) The Attempt at a Solution I've found the dimension for nxn square matrices (n²) and a basis (1 in one place and...

Homework Statement I think I saw another thread answer this question, but I was a little lost whilst reading it. I have just recently learned of the rational root theorem and was using it quite happily; figuring out what possibly answers went with cubic and quartic polynomials gave new...

I have need to calculate the residues of some functions of the form \frac{f(x)}{p(x)} where p(x) is a polynomial. To be more specific I have already calculated the 2 residues of \frac{1}{x^2+a^2}. That one was quite easy. Now I'm asked to calculate the residues of...

POlynomials (or Taylor series ) of the form P(x)= \sum_{n}a_{2n}X^{2n} with a_{2n}\ge 0 strictly have ALWAYS pure imaginary roots ?? it happens with sinh(x)/x cos(x) could someone provide a counterexample ? is there an hypothesis with this name ??

Homework Statement I recently came across this integral while doing a problem in electromagnetism (I'm not sure if there exists a nice analytic answer): \int_{0}^{\pi}P_m(\cos(t))P_n(\cos(t)) \sin^2(t) = \int_{-1}^{1}P_m(x) P_n(x) \sqrt{1-x^2}, Homework Equations P_m(x) is the m^th...

hello everyone:smile: for i=1,2,...,(n+1) let P_{i}(X)=\frac{\prod_{1\leq j\leq n+1,j\neq i}(X-a_j)}{\prod_{1\leq j\leq n+1,j\neq i}(a_i-a_j)} prove that (P_1,P_2,...P_{n+1}) is basis of \mathbb{R}_{n}[X] . i already have an answer but i don't understand some of it. ... we have...

1. Let W be the linear subspace spanned by the polynomials 1 and x. Find an orthogonal projection of the polynomial p(x) = 1+x^2 to W. Find a basis in the space W(perp) My problem is that I don't know how to represent W as a matrix so that I could apply the orthogonal projection formula...

this is really perplexing. how can it be exact? simpsons rule uses quadratics to approximate the curve. how can it be exact if I am approximating a cubic with a quadratic?

trying to show that polynomials f(x), g(x) in Z[x] are relatively prime in Q[x] iff the ideal they generate in Z[x] contains an integer.Thanks .Not homework

Are there any general methods to solve the following complex exponential polynomial without relying on numerical methods? I want to find all possible solutions, not just a single solution. e^(j*m*\theta1) + e^(j*m*\theta2)+e^(j*m*\theta3) + e^(j*m*\theta4) + e^(j*m*\theta5) = 0 where...

In my third year math class we were asked a question to prove that Ho(X) and H1(x) are orthogonal to H2(x), with respect to the weight function e^(-x^2) over the interval negative to positive infinity where Ho(x) = 1 H1(x) = 2x H2(x) = (4x^2) - 2 i know that i have to multiply Ho(x) by...

Homework Statement Define the inner product of two polynomials, f(x) and g(x) to be < f | g > = ∫-11 dx f(x) g(x) Let f(x) = 3 - x +4 x2. Determine the inner products, < f | f1 >, < f | f2 > and < f | f3 >, where f1(x) = 1/2 , f2(x) = 3x/2 and f3(x) = 5(1 - 3 x2)/4 Expressed as a...

I'm trying to show that a set W of polynomials in P2 such that p(1)=0 is a subspace of P2. Then find a basis for W and dim(W). I have already found that the set W is a subspace of P2 because it is closed under addition and scalar multiplication and have showed that. The thing I'm stuck on...

Do Orthogonal Polynomials have always real zeros ?? the idea is , do orthogonal polynomials p_{n} (x) have always REAl zeros ? for example n=2 there is a second order polynomial with 2 real zeros if we consider that there is a self-adjoint operator L so L[p_{n} (x)]= \mu _{n} p_{n} (x)...

this seems more like a physics word problem, I am not even sure how to set this problem up to use it as a polynomial. "A person holds a pistol straight upward and fires. The initial velocity of most bullets is around 1200 ft/sec. The hieght of the bullet is a function of time and is...

Divide the polynomials by using long division. (-9x^6+7x^4-2x^3+5)/(3x^4-2x+1) When I attempted it I started by pulling using 3x^2 . multiplied that by the (3x^4-2x+1) and from there I had to use a fraction of 7/3 or something and then couldn't divide into x cubed. If anyone can...

Homework Statement Find the Taylor Polynomial T2(x) (degree 2) for f(x) expanded about X0. f(x)=3x + cos(3x) X0= 0 Find the error formula and then find the actual (absolute) error using T2(0.6) to approx. f(0.6). The Attempt at a Solution As I've said on this forum before...

Trying to solve a question in linear algebra. P2 is a polynomial space with degree 2. Is P(t): P'(1)=P(2) (P' is the derivative) a subspace of P2. What is the basis ? It seems that it is a subspace with basis 1-t,2-t2. Can anybody explain how this can be found?

Homework Statement If P is the set of all 4-degree polynomials, and W is the subset of all 4-degree polynomials such that p(-2) = p(2), find a set S such that W = span(S). Homework Equations The Attempt at a Solution My guess is that one set that works is x^4, x^2, and 1. My...

Homework Statement A chare +Q is distributed uniformly along the z axis from z=-a to z=+a. Find the multipole expansion. Homework Equations Here rho has been changed to lambda, which is just Q/2a and d^3r to dz. The Attempt at a Solution I have solved the problem correctly...

Homework Statement I need to simplify this: ((84/13)x4y - 4) / (-x + (21/13)x5)y)Homework Equations The Attempt at a Solution I don't know if it can be simplified further. I can't factor anything out that will cancel. I multiplied both the top and the bottom by 13 to get rid of those fractions...

Polynomials divisible by... Homework Statement 1) Explain why n^3 - n is always divisible by 3 for any n that is an element of the natural numbers. 2) Give 2 other polynomials that are always divisble by 3. 3) Give 2 polynomials that are divisible by 2 but not 4, and 2 other polynomials...

Homework Statement A polynomial p(x) leaves the rest 3 when divided by (x+2) and the rest 8 when divided by (x-6). What's the rest r(x) when p(x) is divided by (x+2)(x-6)? Homework Equations The Attempt at a Solution I wrote the three equations: p(x)=q1(x+2) + 3 p(x)=q2(x-6) +...

Homework Statement Consider the polynomial p(x)=x^6-1. (Apply over any field F). (a) Find two elements a,b \in F so that p(a)=p(b)=0. Then use your answer to find two linear factors of p(x). (b) Show that the other factor of p(x) is x^4+x^2+1 (c) Verify the identity...

I see in MATLAB that you can call legendre(n,X) and it returns the associated legendre polynomials. All I need is is the simple Legendre polynomial of degrees 0-299, which corresponds to the first element in the array that this function returns. I don't want to call this function and get this...

Hi I have a set of nonlinear equations f_i(x_1,x_2,x_3...) and I want to find their solutions. After doing some reading I have come across commutative algebra. So to simplify my problem I have converted my nonlinear equations into a set of polynomials p_i(x_1,x_2,x_3...,y_1,y_2...) by...

Homework Statement Let P(n,m) be the space of all polynomials z with complex coefficients, in two variables s and t, such that either z = 0 or else the degree of z(s, t) is <= m - 1 for each fixed s and <= n - 1 for each fixed t. Prove that there exists an isomorphism between Pn (x)...

Homework Statement (x_1 - x_2)(x_3 - x_4) Find permutations of subscripts that leave value unchanged Homework Equations The Attempt at a Solution Okay so I know that it's asking how I should rearrange things and still not change the value. Switching 1 with 2 or 3 with 4 would work but I know...

There's a question in Calculus by Spivak about polynomials and I was wondering about how to construct them to have specific roots or values at certain points. For example it says if x_{1}, ..., x_{n} are distinct numbers, find a polynomial f_{i} such that it's of degree n-1 which is 1 at x_{i}...

I've been trying to work out a bunch of problems that have to do with finding irreducible polynomials, and this one really seemed to stump me... What is the probability that a random monic polynomial over F_3 of degree exactly 10 factors into a product of polynomials of degree less than or...

HI there, I have a tiny question concerning the gcd of polynomials. Assume, \chi is the greatest common divisor of the polynomails p_{ij}, i,j=1,2. I then form q_{11}=p_{11}^2+p_{12}^2,\quadd q_{12}=p_{11}p_{21}+p_{12}p_{22},\quadd q_{21}=p_{21}p_{11}+p_{22}p_{12},\quadd...

Very Interesting Question on Division of Polynomials! [b]1. Question: 'When a polynomial f(x)= x^4 - 6x^3 + 16x^2 - 25x + 10 is divided by another polynomiall g(x)= x^2 - 2x + k, the remainder is x+a. Find the value of a k and a'. Homework Equations [b]3. I tried solving it by...

Here is something that always bugged me, and I think I have an explanation for it now, but I am wondering if it is correct. Alright, the problem to me was that back when I was in Diff-eq, to use undetermined coefficients with polynomials, we would always group together the terms on one side, and...

Homework Statement A is a square matrix of size n, B is of size m, C is an m*n(typo,should be n*m) matrix and n>m ,Rank(C)=m. if AC=CB, prove characteristic polynomial of B divides that of A. Homework Equations nothingThe Attempt at a Solution I think I need to prove any eigenvalue of B is an...

Homework Statement Let S denote the collection of all polynomials of the form p(t) = (2a - b)t^2 + 3(c - b)t + (a - c), where a,b,c are real numbers. Determine whether or not S is a subspace of P2. The Attempt at a Solution Okay, so I know that in order for S to be a subspace, it must...

Question: Can someone explain the following to me: "... the entries of the matrix A - tI_n are not scalars in the field F. They are, however, scalars in another field F(t), the field of quotients of polynomials in t with coefficients from F." I asked someone earlier today, and I got an...

Homework Statement Let P3 be the space of all polynomials (with real coefficients) of degree at most 3. Let D : P3 -> P3 be the linear transformation given by taking the derivative of a polynomial. That is D(a + bx + cx2 + dx3) = b + 2cx + 3dx2: Let B be the standard basis {1; x; x2; x3}...

Homework Statement Show the the equation f(z) = z^4 + iz^2 + 2 = 0 has two roots with |z|=1 and two roots with |z|=sqrt(2), without actually solving the equation.Homework Equations Rouche's theorem, the argument principle?The Attempt at a Solution This is what I have done so far: First show...

Hello, I'm working through Polynomials by Barbeau. I think I may have provided a decent proof for one of the exercises, but I'd like a second opinion. Here's the exercise: Show that every quadratic equation with complex coefficients has at least one complex root, and therefore can be written...

Please somebody help me with this it is very urgent. I have that f(x) = x^5-5x+1 has S_5 as galois group over rationals. ANd M is the splitting field of f(x) over rationals. Then how can I show that : determine f(x) is irreducible over Q({-51}^{1/2})[x] or not? Determine if there is...

I am having trouble evaluating the Legendre Polynomials (LPs). I can do it by Rodrigues' formula but I am trying to understand how they come about. Basically I have been reading Mary L. Boas' Mathematical Methods in the Physical Sciences, 3E. Ch.12 §2 Legendre Polynomials pg566. In the...

Hi, there are a few questions and concepts I am struggling with. The first question comes in 3 parts. The second question is a proof. Question 1: Please Click on the link below :smile: Question 2: Please Click on the link below :smile: For Q2, could you please show me how to...

Homework Statement Find a polynomial with integer coefficient for which \sqrt(2) + i is a zero. Homework Equations The Attempt at a Solution I'm not sure where to really start with this one. It is on my review sheet, and I can't remember how to do it. Could someone give me a hand?

My book reads as follows: "This is an entirely elementary algebraic formula concerning a polynomial in x or order n, say f(x) = a_0 + a_1x + a_2x^2 + ... + a_nx^n . If we replace x by a + h = b and expand each term in powers of h, there results immediately a representation of the form...