Polynomial Definition and 1000 Threads

Let [ tex ]f(x)=\sum_{i=0}^n c_i x^i[ / tex ] be an arbitrary polynomial function of degree n Show that if f(0)=0 then either f is constant or f(x)=xg(x), where g is a polynomial function of degree n-1 I don't know how to start. Please help Thank you in advance

Dear All Friends, I am currently working on a project which needs some orthogonality integration formulae of Laguerre polynomials. I referred worlfram's math function site http://functions.wolfram.com/Polynomials/LaguerreL3/21/02/01/ and get three seemingly useful ones...

Homework Statement Let p be a polynomial. Show that the roots of p' are real if the roots of p are real. Homework Equations The Attempt at a Solution So we start with a root of p', call it r. We want to show that r is real. Judging by the condition given, I am assuming that...

Homework Statement Give an example of a polynomial irreducible in Q[x], but reducible in Z[x] Homework Equations The Attempt at a Solution I think there is no example of this. The coefficients of a reducible polynomial with integer coefficients will be rational, or am I mistaken?

Homework Statement Show that for any square matrices of the same size, A, B, that AB and BA have the same characteristic polynomial. Homework Equations The Attempt at a Solution I understand how to do this if either A or B is invertible, since they would be similar then. I saw a...

I have a small idea on what irreducible and primitive polynomials are in Abstract algebra. But what is minimal polynomial? -Devanand T

Homework Statement Hi An integration question: t= 1/(1+r2) Can you please show me how to integrate with respect to r. Thank you. The Attempt at a Solution ∴t = (1+r2)-1 I then tried substituting u = 1+r^2 but that didn't work! Is there a trick with this?

I have a quick question. The problem reads: Prove that there is no integer m such that 3x2+4x + m is a factor of 6x4+50 in Z[x]. Now, Z[x] is not a field. So, the division algorithm for polynomials does not guarantee us a quotient and remainder. When I tried dividing 6x4+50 by 3x2+4x +...

Homework Statement Applying remainder theorem again and again to show that the remainder of the f(x) polynomial function when divided by (x-α)(x-β) is A(x-α)+B . Determine A and B Homework Equations the remainder of a polynomial f(x), divided by a linear divisor x-a, is equal to f(a) The...

Can I use excel to make an equation for 4 variables (x,y,z,w) e.g a,b,c,d,3,f,g,h,i, would be constants w = ax + by + cz + dx^2 + ey^2 + fz^2 + gxy + hyz + iyz What are these equations called ? What literature can I study to better understand the procedure of making these tye of...

i need the derivation of orthogonal properties of associated laguerre polynomial (with intermediate steps). someone please tell me where can i get it (for easy understanding).

I understand that mathematicians have had to define the number '0' also as a polynomial because it acts as the additive identity for the additive group of poly's.What I do not understand is why they define the degree of the zero polynomial as [ tex ]-\infty[ /tex ]. An explanation on planetMath...

Background information: I have come up against a mathematical question which I as somebody with relatively limited exposure to maths can not seem to answer. I am a student working on a thesis dealing with near-infrared spectroscopy (NIRS). The NIR scanner is able to measure the moisture content...

Let's say I have the equation f(x) = 2x + 3 * (3x^2 + 3) - x^2 + 5. If my algebra is right, this is a 3rd-degree polynomial. How many zeroes does this equation have? How did you figure that out?

Homework Statement My teacher likes to teach us the D-notation methods for higher order DE's. I am having a hard time with this one and I can't seem to find the formula for the general solution Find a fundamental set of the equation (D-1)^{2}(D^{2}-6D+13)^{3}y = 0 Homework...

Homework Statement What are the steps to factoring 3rd order polynomials like x^3+8x^2-21x+10? It's to find eigenvalues of a matrix in linear algebra, I completely forgot how to factor and it's killing me. Homework Equations The Attempt at a Solution None, unless its a polynomial...

I couldn't get these lines from my book. I will reproduce it here. Warning: In step 1, if you use computer to fit a polynomial to the data , it could lead to disaster. For example, consider fitting a sixth degree polynomial to the seven data points, or, an (n-1) degree polynomial to n...

factor completely 2x3 - 8a2x + 24x2 + 72x so far i got 2x(x2 + 12x + 36) - 8a2x 2x(x+6)(x+6) - 8a2x don't know what to do from here.

Unless I'm missing something here, I've noticed that if you want to store a polynomial function on the TI-83 or the TI-84 Plus, you have to create a program that asks you what the value of x is, then displays the value of f(x). I kind of wish I could define a function without making a program.

I'd like to know how to find the root of a polynomial on my TI-84 Plus without this "Polynomial Root Finder and Simultaneous Equation Solver" app. The reason is that the app's not in my calculator and I can't transfer the app to my calculator. I keep getting an "Access Denied" error message...

The problem is :- Integral of (1+x^4) / ( 1 + x^6) . dx I have reduced it to a form of Integral of 2.sqrt(tantheta) / ( 1 + tan^3 theta ) over dtheta where x^2 = tantheta. However I cannot reduce it further. How do I proceed ? In general, how do I proceed given a problem of the form...

Are the roots of a polynomial given by the function f(x) defined as the values for x where f(x)=0? Does that mean f(x)=x^2 has only one root? Even though for every other value of x except zero there are two values for x that you can input to output a particular value for f(x). What about...

Homework Statement (A somewhat similar question to my last one). Let J be the ideal of the polynomial ring \mathbb{Q}[x] generated by x^2 + x + 3. Find the multiplicative inverse of (3x^3 + 3x^2 + 2x -1) + J in \mathbb{Q}[x]/JHomework Equations The Attempt at a Solution I think I need to apply...

Homework Statement Factorize x^2 + x + 8 in \mathbb{Z}_{10}[x] in two different waysHomework Equations The Attempt at a Solution I can see that x = 8 = -2 and x = 1 = -9 are roots of the polynomial, so one factorization is (x + 2)(x + 9). Is there a systematic way to find all the factorizations?

Find an approximate value of the number e-0.1 with an error less than 10-3 ı know that ex = Ʃ(from zero to ınfinity) xn / n!=1+x/1!+x 2/2!+... ı don't know how to use e-0.1 in this question.Do ı write -0.1 instead of x in ex series?

Homework Statement If α, β and γ are roots of cubic polynomial and: αβγ = 6 α2+β2+γ2=20 α3+β3+γ3=121 Find the equation of cubic polynomial Homework Equations vieta The Attempt at a Solution The equation is in the form: x3 - (α+β+γ)x2 + (αβ + αγ + βγ) x - αβγ = 0 But I...

Now if I have a function y=(ab+ac)/a, it can be further factorised, y=(a(b+c))/a. Now if we cancel off the a, we will have only y=b+c that will also give the same y-values as the original form of the function y with respect to the same x-value. This statement implies that cancellation or...

Homework Statement For any quadratic polynomial ax2+bx+c having zeros β and α Prove that β + α = -b/a and αβ = c/a. Homework Equations The Attempt at a Solution I have found a method myself to prove α+ β = -b/a. However, I could not prove αβ = c/a. It goes like this. If α and β are the...

I want to find the root(for N) of this equation: \frac{(2N-1)^2}{N(1-N)}=Ce^t The hint says "consider taking a substitution u=N-1/2" ...which is the top bit of the fraction. But what does take a substitution here mean ? This is a part of a loooong modelling problem which involved an ugly...

"Given operators σ,τ on a finite-dimensional space V, show that στ=i, and that σ=p(τ) for some polynomial p in F[x]." The first part was no problem. As for the second, I have a strong suspicion that p is the characteristic polynomial, mostly because I believe I heard of that fact before (that...

Hello, I just want to clear up a confusion. Is f(x,y) = a + bx + cy + dxy a quadratic polynomial? where x and y are variables and a,b,c,d are constants.

This is the end of a triple integration problem. I can get down to what seems like it should be a simple polynomial integral of a single variable. Yet I just can't get the numbers to work out. \int_0^5 \frac{-1}{2}x^3 + \frac{15}{2} x^2 - \frac{75}{2} x + \frac{125}{2} dx The indefinite...

I created a method for both approximating a function and extending a it's domain from a Natural to a Real Domain. Does this have a name already or any interesting application? Basically. Add polynomial of degree 0, 1, 2, 3, etc. Making at the same time the approximation function equal to f(0)...

Homework Statement Find the first 3 non-zero terms of the Taylor polynomial generated by f (x) = x^{3} sin(x) at a = 0. Homework Equations f^{n}(x) * (x-a)^{n} / (n!) The Attempt at a Solution I got the question wrong: my answer was 1/3! + 1/5! + 1/7! Here is the answer below. I...

If $p(x)$ be a polynomial of degree $98$ such that $\displaystyle p(x) = \frac{1}{x}$ for $x=1,2,3,...,98$ Then value of $p(100)=$

How would I go about approaching this problem? Given the polynomial: x^100 - 3x + 2 = 0 Find the sum 1 + x + x^2 + ... + x^99 for each possible value of x.

How would one evaluate $$\Phi = \int_{-\infty}^{+\infty} e^{-(ax+bx^2)} dx$$. I was trying to change it into a product of an error function and a gamma function, but I needed an extra dx. Any other ideas?

[FONT=Times New Roman]Let a,b,c,d be real numbers. Sauppose that all the roots of the equation $z^4 + az^3 + bz^2 + cz + d = 0$ are complex numbers lying on the circle $\mid z\mid = 1$ in the complex plane. The sum of the reciprocals of the roots is necessarily: options a) a b) b c) -c d)...

In the Turing Machine, the machine accepts a word if the computation terminates in the accepting state. The language accepted by the machine, L(M), has associated an alphabet Δ and is defined by L(M) = {w \in \Delta} This means that the machine understands the word w if w belongs to the...

If $F$ is the field of rational numbers, find the necessary and sufficient conditions on $a$ and $b$ so that the splitting field of $p(x)=x^3+ax+b=0$ has degree exactly $3$ over $F$. ATTEMPT: If $p(x)$ is not irreducible in $F[x]$ then the splitting field of $p(x)$ over $F$ can have degree...

Homework Statement p(x)=((x−1)^2 −2)^2 +3. From here find the full factorization of p(x) into the product of first order terms and identify all the complex roots. Homework Equations I am having trouble doing this by hand. I know there are four complex roots but can't seem to figure out...

Polynomial function f(x)= x^3-12x^2+46x-52 A. List possible rational ZerosB. find all the zeros (real and complex) of the function (test x=2 as a rational zero using the synthetic division?

Can anyone explain ideals and polynomial rings i.e. definitions, examples, the most important theorems, etc.?

[FONT=Times New Roman]If p(x) is a polynomial such that p(0)=5 ,p(1)=4 ,p(2)=9,p(3)=20 , the minimum degree it can have

Hi guys, I'm not entirely good with factoring so I was wondering if any of you can show me how I would go about factoring this polynomial: (With necessary steps if you can Please and thanks!) 36mx + 10y - 24x - 15my

we assume a,b,c,d are unknown variables,whose solution are either 0 or 1.So a power of a variable equals to itself(e.g,a^(n)=a).Would u please help me find a proper way to sovle the following simultaneous equations whose solutions are either 0 or 1...

Is it possible to have a cubic polynomial (ax^3+bx^2+cx+d) which has three REAL roots, with one of them being +/- infinity? If there is, can you give an example? Thanks!

Homework Statement x^2-x-13/(x^2+7)(x-2) hello i am having trouble solving this problem.. could anyone please show me how to do this step by step? i know polynomial long division is required before it can be converted to partial fractions. I also know the answer is 2x+3/x^2+7 - 1/x-2...

Hi, I've been working on this question which asks to show that {{P}_{n}}(x)=\frac{1}{{{2}^{n}}n!}\frac{{{d}^{n}}}{d{{x}^{n}}}{{\left( {{x}^{2}}-1 \right)}^{n}} So first taking the n derivatives of the binomial expansions of (x2-1)n...

Hi guys, I've been working on a question which is as follows: For which real values of c will the set $\{1+cx, 1+cx^2, x-x^2\}$ be a basis for $P_2$? I'm coming up with the answer as no values of c, but am I really wrong? I've only checked linear independence, because it would imply that it...