Polynomial Definition and 1000 Threads

Using Matlab, I have 4 polynomial equations which I can solve by substituting the starting values for the variables into the equations. However, I want to contruct a loop to then substitute the answer from the first round back into the equations to replace the orginal starting values, and do...

Is there a function in Matlab which gives a list of numerical approximations to the roots of a polynomial equation (eg. like NSolve in Mathematica?) Thanks!

Homework Statement Solve: x^3 - 9x^2 + 15x + 30 Homework Equations The Attempt at a Solution The factors of 30 are +-1, +-2, +-3, +-5, +-6, +-10, +-15, and +-30. I used my graphing calculator and got a zero close to -1. I plugged it into the original equation and got 5, not...

decompossing a polynomial (keep getting the wrong result :() Homework Statement Given the polynomial f(x) = -x^2 - x + 1, decompose the polynomial into linear terms The Attempt at a Solution I get (x-(-(\frac{-\sqrt{5}+1}{2}))((x-(-\frac{-\sqrt{5}-1}{2})) I seem to be missing a...

Again, in my quest to learn things I won't use in a class for at least a year, I've been looking at convolutions. Specifically, after finishing the multiple choice section of an AP Chemistry test 50 minutes early, I looked at the convolution of a polynomial with itself. I'm confused about one...

Homework Statement Let p(z) = \sum_{j=0}^{n} a_{n-j}z^j be a polynomial of at least degree 1 thus n \geq 1. Show that if z\neq 0 then 1/z is a root of the polynomial p. Homework Equations Fundamental theorem of Algebra The Attempt at a Solution If a expand the polynomial...

Homework Statement The curves are: y = \frac{x^{4}}{x^{2}+1} and y = \frac{1}{x^{2}+1} The Attempt at a Solution So again I assume that: \frac {x^{4}}{x^{2}+1} = \frac {1}{x^{2}+1} and then cross multiply: (x^{2}+1) = x^{4}(x^{2}+1) not really sure at this point if i should...

Homework Statement Show that g(x) = x^3 + 1 divides f(x) = x^{9999} +1. Homework Equations The Attempt at a Solution g(x) can obviously be factored into the irreducible polynomials (x+1)(x^2 - x + 1) in Z[x], and since f(-1) = (-1)^{9999} + 1 = 0, the factor theorem gives that (x+1) divides...

Is there any criterion for the irreducibility of a polynomial in several variables over an algebraically closed field (or specifically for the complex numbers)? For one variable, we know this is simply that only degree one polynomials are irreducible, is there anything similar for several variables?

Homework Statement find the taylor polynomial f4 for f(x)=sin(2x) and a=pi/4 Homework Equations sin(x)=((-1)^n)(x^(2n+1))/((2n+1)!) The Attempt at a Solution so replace x with 2x? you get ((-1)^n)(2x)^(2n+1)/(2n+1)!) is this right?

Homework Statement Obtain the Taylor polynomials Tnf(x) as indicated. In each case, it is understood that f(x) is defined for a11 x for which f(x) is meaningful. Problem one Tn = (a^x) = sigma from k = 0 to n of ((log a)^k)/k! x^k Problem two Tn = (1/(1+x)) = sigma from k = o...

Homework Statement Prove the following when p is a positive integer: b^p - a^p = (b-a)(b^{p-1}+b^{p-2}a+b^{p-3}a^2+...+ba^{p-2}+a^{p-1}) Hint: Use the telescoping property for sums. Homework Equations None The Attempt at a Solution I tried reducing it to, (b-a)\sum_{k=1}^p...

Homework Statement Consider the data in the following table for constant-pressure specific heat, C p (kJ/kg.K) at various temperatures T (K). Determine the simplest interpolating polynomial that is likely to predict Cp within 1% error over the specified range of temperature. T : 1000 1100...

Homework Statement In the polynomial x^5+22x^3-34x^2+117x-306 given that the roots on the real and imaginary plane are all integers, factorize the polynomial into real linear and quadratic factors. The Attempt at a Solution I was able to find the real factor, which is x=2 and then I...

Homework Statement Derive a method for approximating f'''(x0) whose error term is of order h^{2} by expanding the function f in a fourth taylor polynomial about x0 and evaluating at x_{0} \pm h and x_{0} \pm 2h. Homework Equations The Attempt at a Solution I'm not sure where to...

Homework Statement Let P(x) be any polynomial of degree at least 2, all of whose roots are real and distinct. Prove that all of the roots of P'(x) must be real. What happens if some of the roots of P are multiple roots? Homework Equations I think that question is related to the concept...

Homework Statement The question asks me to write out a taylor polynomial for 1/(1-x^2) of degree 2n+1 at 0. The Attempt at a Solution My answer was 1 + x^2 + x^4 + x^6 + ... + (x^4)/(1-x^2) which I just got from using hte geometric series formula. The textbook answer however...

Homework Statement The first 3 parts of this 4 part problem were to derive the first 5 Hermite polynomials (thanks vela), The first 5 Legendre polynomials, and the first 5 Laguerre polynomials. Here is the last part: Write the polynomial 2x^4-x^3+3x^2+5x+2 in terms of each of the sets of...

Let f, g \in \mathbb{Z}[x, y, z] be given as follows: f = x^8 + y^8 + z^6 and g = x^3 +y^3 + z^3. Express if possible f and g as a polynomial in elementary symmetric polynomials in x, y, z. Professor claims there is an algorithm we were supposed to know for this question on the midterm. I...

Homework Statement Let f(x)=a0+a1x+a2x2+a3x3+a4x4. Show that if a0a4<0, then f(x)=0 have at least 2 real solutions. Homework Equations The Attempt at a Solution Any hints? I can't tell how to begin an attempt for a solution.

Homework Statement If z = cosθ + isinθ, show that zn + z-n = 2cosnθ. Hence solve the equation 3z4 - z3 + 4z2 - z + 3 = 0 for complex z. (Hint: Reduce the given equation to an equation with the unknown cosθ) Homework Equations z = cosθ + isinθ zn + z-n = 2cos(nθ) 3z4 - z3 + 4z2 - z + 3...

Hi everybody: I'm trying to analyse equations of analogue electronic filters, but i can't get the "norm" of the complex polynom if i use variables (or symbols? what's the difference?), here's the simplest exemple (RC low pass filter, a represents R*C*w, reals variables > 0): Assuming[ a...

Homework Statement Determine the order two Taylor polynomial, p2(x, y), for f(x, y) = log e (1 + x2 + y4) about point (0, 1) ANSWER: loge (2) + 2y - 2 + \frac{1}{2} [ x2 - 2y2 + 4y - 2 ] Managed that question and should be correct. If not, do let me know =) Part 2: Using...

Hello! Trying to measure the magnetic field of the Earth here. Basically, measured the deflection of the current carrying wire in the vertical plane. However, the sag of the wire due to heat overwhelmed the displacement due to the magnetic field. If I plot displacement of wire against current, I...

Homework Statement Find any extrema, points of inflection, asymptotes, and symmetry for function. Homework Equations f(x) = (x^5-10x^3+9x) / ( x^4 - 16) The Attempt at a Solution Extrema: I took the first derivative by using the Quotient Rule, and got (x^8 + 10x^6 - 107x^4 +...

The function y=e^x can be expanded using the power series, thus y=e^x=1+x+\frac{x^2}{2}+\frac{x^3}{3!}+... This is a polynomial of infinite degree, and the theorem that says a polynomial must have at least one root in the complex field, and thus this extends to a polynomial of nth degree having...

Problem: Let R=Q[y] and suppose f,g \in F[x] both have degree 10 with respect to x and degree 6 with respect to y. Suppose h = gcd(f,g) has degree 4 with respect to x and degree 2 with respect to y. Derive an upper bound (as good as possible) on the number of distinct integers i such that...

Homework Statement One root of 2x2-kx+k=0 is twice the other. Find K. (assume k is not 0) Homework Equations not sure what this means. The Attempt at a Solution \alpha+\beta=\frac{-b}{a} \alpha\beta=\frac{c}{a} then i tried \alpha=2\beta then substituted in...

Homework Statement Let A and B be to nXn matrices and A is invertible. Prove: P_{AB}(x)=P_{BA}(x) The Attempt at a Solution Since A is invertible we have det(A)=0. det(AB)= det(A)det(B)=0det(B)=0 ->P_{AB}(x)=P_{BA}(x)=0 Is that correct? [

I just need help on how to start the problem, I'm not asking anyone to do it for me, I'm just slightly confused. What is the degree of the Taylor polynomial needed to approximate sqrt(e) with error < 0.001. Use ex as your function, with x = 0.5. I'm just honestly confused on where to even...

Hi I am presented with the following problem Homework Statement Let P_{4}(\mathbb{R}) = \{a_0 + a_1 \cdot x + a_2 \cdot x^2 + a_3 \cdot x^3|a_0,a_1,a_2,a_3 \in \mathbb{R} \} Then let S be a subset of P_4(\mathbb{R}) where S = \{a_0 + a_1 \cdot x + a_2 \cdot x^2 + a_3 \cdot...

Homework Statement Which one of the following subsets of P_{2} (degree of 2 or below) are subspaces? a) a_{2}t^{2} + a_{1}t + a_{0}, where a_{1} = 0 and a_{0} = 0 b) a_{2}t^{2} + a_{1}t + a_{0}, where a_{1} = 2a_{0} c) a_{2}t^{2} + a_{1}t + a_{0}, where a_{2} + a_{1} + a_{0} = 2 Homework...

Homework Statement I'm working on a problem which is show that the matrix A = (^{a}_{c} ^{b}_{d}) is diagonalisable iff (a-d)2 + 4bc > 0 or a-d = b = c = 0 Homework Equations The Attempt at a Solution I've shown if (a-d)2 + 4bc > 0 then the characteristic polynomial has 2...

Homework Statement I am unsure what to do here. I'm given the equation 5(x+1)3 = -5 , and I was given the equation 5(x+1)3 = 0 beforehand. The latter equation has an 'x' value of -1 only, right. However, for the former equation, because I brought the '-5' across the equal sign I realize the...

Homework Statement A is an nxn square, real matrix. Let f(x) be the characteristic polynomial, write f(x) = xn - c1xn-1 + ... + (-1)rcrxn-r + ... + (-1)ncn Show that cn-1 = \sum det (Aii) where Aii is the (i,i) minor of A. Similarly, what is the coefficient cr? Homework Equations...

This is from my Vector Analysis course today. We've been doing a bit of abstract stuff since class began, but the professor said we're going to get to concrete stuff pretty quickly. I think the notation is throwing me off a bit. I'm not sure why the alphas change for each successive...

[SIZE="5"]Hi, I'm in Engineering Foundation. I'm stuck in one limit question. Find the limit :_ **********________ Lim (3x + V 9x^2 - x ) x-> -infinity by substitution it gives ( inf - inf ) I tried to solve it and get -inf Can anyone help me please ?

Homework Statement E1 = {pn(t) = nt(1-t)n:n in N}; E2 = {pn(t) = t + (1/2)t2 +...+(1/n)tn: n in N}; where N is set of natural numbers is the polynomial bounded w.r.t the supremum norm on P[0,1]? Homework Equations supremum norm = ||*|| = sup{|pn(t)|: t in [0,1]} The Attempt...

Hi, I have a question, which seems deceptively simple to me, but when I thought about it, I couldn't really come up with a rigorous proof. Here goes, Are the roots of a polynomial equation unique? Suppose we have a general monic polynomial equation: z^{n} + c_{1}z^{n-1} + c_{2}z^{n-2} +...

Homework Statement 7X2(X - 2) - (X - 2) 2. The attempt at a solution Ok, this is really simple but for some reason I can't figure out why the answer has to be (7X2 - 1)(X - 2) The process is that I have to divide both parts by (X - 2). However, if I do that, I only get 7X2 - 1...

Homework Statement Not so much a homework problem, but a problem that is annoying me because of its simplicity. Not all cubic polynomials with rational coefficients can be factorized by the rational root theorem (or is this false?). What I am finding hard to comprehend is how a cubic with...

Homework Statement In solving a question I got a problem of solving the following integral. Your comments are appreciated. Homework Equations \int_{-1}^{1}xP_l'(x)dx=? The Attempt at a Solution I tried to solve by integration by parts, i.e...

Homework Statement Determine the nth order Maclaurin polynomial for 1/(1-x)2 Homework Equations The known Maclaurin polynomial series: 1/(1-x)= 1+x+x2+x3+...+xn +O(xn+1) The Attempt at a Solution I tried expanding the bottom to get 1/ (1-2x+x2)) then wrote it in the form to match...

Homework Statement Find the Maclaurin polynomial for P(6)X for 1/(1+2x2) about x=0 Homework Equations I'm pretty sure I need to use the Maclaurin polynomial for 1/(1-x)= 1+x+x2+x3+...+xn +O(xn+1)The Attempt at a Solution I simply rewrote as 1/(1-(-2x2)) then proceeded to sub and expanded...

Homework Statement Show 16x^4 = 8x^3 - 16x^2 - 8x + 1 is irreducible. Homework Equations Eisenstein's criteria, if there is n s.t. n does not divide the leading coefficient, divides all the other coefficients, and n^2 does not divide the last coefficient then the polynomial is...

Homework Statement A is an nxn invertible matrix. Show that PA-1(x) = (\frac{(-x)^n}{det(A)}) *PA(1/X) Homework Equations det(AB) = det(A)det(B) The Attempt at a Solution Ok from what I understand this is the proof for the characteristic polynomial for the inverse of a...

Dear All, I am implementing the scattering of dielectric sphere under electromagnetic plane wave. The expression of the field contain \frac{P_n^1(\cos\theta)}{\sin\theta} and\sin\theta P_n^1'(\cos\theta), where the derivative is with respect to the argument. These two terms are giving me...

Homework Statement f(x) = \frac{ln(3x)}{6x}, a = \frac{1}{3}, n=3 Find T3 Homework Equations Taylor Series - f(n)(x)/n! * (x-a)^n The Attempt at a Solution So, I isolated ln(3x) from 1/6x. I created the series based off of ln(3x). f(0)(x)=ln(3x) ->f(0)(1/3)=ln(3(1/3)) =0...

Homework Statement Basically, what is : \left. {\left[ {\ln (1 - x) + \ln (1 + x))(P_{l - 1} (x) - xP_l (x)} \right]} \right|_{x = - 1}^{x = 1} that is, the limits of that function as x->1 and -1. P_l(x) is the legendre polynomial of the first kind of order l. Homework Equations \[ P_l (...

Homework Statement Prove that (ax^n)' = nax^n-1 using induction. I am very weak with induction proof, and I haven't had much trouble proving the basis step, but I can't seem to finish it... Homework Equations The Attempt at a Solution 1. Prove (ax)' = a (a(x+h) - a(x))/h =...