Remainder Definition and 171 Threads

Find the value of k for which (a-3b) is a factor of a4 - 7a2b2 + kb4. Hence, for this value of k, factorize a4 - 7a2b2 + kb4 completely. I tried to do it but my mind is not going anywhere. Any help will be greatly appreciated. :)

I'm struggling with how to even begin with this problem. Find the remainder of the division of 75!*130! by 211. 211 is prime, so I know the remainder is not 0. I'm not sure where to start though. Thanks!

Number Theory -- Find Remainder .. when dividing by 17 Homework Statement Find the remainder when 3^24*5^13 is divided by 17. Homework Equations I know that 3^24 = 16 (mod 17) and calculated that 5^13 mod 17 = 3 (mod 17) The Attempt at a Solution BUT, I'm completely unsure...

Homework Statement Find the Taylor polynomial for f(x) = 1/(1-x), n = 5, centered around 0. Give an estimate of its remainder. The Attempt at a Solution I found the polynomial to be 1 + x + x2 + x3 + x4 + x5, and then tried to take the Lagrange form of the remainder, say, for x in [-1/2, 1/2]...

Is there anywhere in topology where one would see the Chinese Remainder Theorem?

Homework Statement (For power series about x=1) Using the error formula, show that \left|ln(1.5)-p_{3}(1.5)\right|\leq\frac{(0.5)^{4}}{4} Homework Equations p_{3}(x) = x-1 - \frac{(x-1)^{2}}{2} + \frac{(x-1)^{3}}{3} \\\epsilon_{n}(x)=\frac{f^{n+1}(\xi)}{(n+1)!}(x-x_{o})^{n+1}\\where \xi lies...

Homework Statement The remainder theorem can't really be applied when dividing by something other than a linear equation since you wouldn't know what a is, right? Homework Equations The Attempt at a Solution

So I'm studying for a final, and it just so happens my professor threw taylor polynomials at us in the last week.. I understand the concept of a taylor polynomial but i need some help fully understand the LaGrange remainder theorem if we have a function that has n derivatives on the interval...

Homework Statement Please assume that what I have for the remainder is correct, and we are on the domain 2 < x < 4 around 2. Homework Equations The Attempt at a Solution 0 \leq |R_{n} (2,x)| = \frac{1}{n+1} |\frac{x-2}{z_{n}}|^{n+1} Since, 2 < x < 4 then, 0 < x-2 < 2...

Homework Statement Use Taylor's remainder formula to show that the Taylor series for f(x) is about the point indicated converges to f(x) for all x. f(x) = e^{5x} about x=0 Homework Equations The Attempt at a Solution Since, f^{n}(x) = 5^{n}e^{5x}, Taylor's remainder...

The Chinese remainder theorem tells us that the system of equations: \begin{align} x &\equiv a_1 \pmod{n_1} \\ x &\equiv a_2 \pmod{n_2} \\ &\vdots \\ x &\equiv a_k \pmod{n_k} \end{align} Uniquely determines all numbers in the range: X<N=n_1n_2\ldots n_k and that all solutions are...

Homework Statement How much is the balance if u want to get integer value of ((2^1000) divided by 7)) Homework Equations The Attempt at a Solution I need a little hint to start off with an attempt.

Homework Statement Find the indicated value of the polynomial using the Remainder Theorem p(x)=2x^3-2x^2+11x-100; find p(3) Homework Equations p(x)=2x^3-2x^2+11x-100 The Attempt at a Solution Synthetic division 3] 2 -2 11 -100 6 12 69 2 4 23 [-31 answer: p(3)=-31 im not...

Hello, I was wondering if anyone could explain to me the thought process behind how you find the maximum remainder of a Taylor series? I read the wiki article and didn't help me at all, http://en.wikipedia.org/wiki/Taylor's_theorem My book talks about something like this(image is...

Homework Statement Okay, well this was a question on one of my recent tests: How many terms do you have to use to estimate the sum from n = 0 to n = infinity of (-e/pi)^n with an error of less than .001? Homework Equations Alternating series remainder theorem: For an...

So I am working on solving sets of linear congruence with the chinese remainder theorem. When I go to solve for the inverses I am meeting a bit of trouble. What do I do when the a term is larger that m? Example 77x=1(mod3) 33y=1(mod7) 21z=1(mod11) where x,y,z are the inverses I am trying...

(a) Let R and S be rings with groups of units R∗ and S ∗ respectively. Prove that (R × S)∗ = R∗ × S ∗ . (b) Prove that the group of units of Zn consists of all cosets of k with k coprime to n. Denote the order of (Zn )∗ by φ(n); this is Euler’s φ-function. (c) Now suppose that m and n are...

Homework Statement Prove that 10^n leaves remainder 1 after dividing by 9. The Attempt at a Solution There is an integer K, such that 10^n = 9k + 1 Where do i go from here if I want to do it just directly?

I'm currently studying the Taylor series and I cannot figure out how the remainder term came to be. If anyone could clarify this for me, I would be really grateful ...! I understand that the Taylor series isn't always equal to f(x) for each x, so we put Rn at the end as the remainder term...

Homework Statement (a)Use Definition 10.8.1 to find the Maclaurin series for f(x) = sinh x. Express your answer using Σ notation. (b) Find the interval of convergence for the series found in part (a). (c) Use the Remainder Theorems 10.7.4 and 10.9.2 to show that the series found in part (a)...

Hello, I am looking into proving that the Chinese Remainder Theorem will work for two pairs of congruences IFF a congruent to b modulo(gcd(n,m)) for x congruent to a mod(n) and x congruent to b mod(m). I have gotten one direction, that given a solution to the congruences mod(m*n), then a...

Homework Statement What degree Taylor Polynomial around a = 0(MacLaurin) is needed to approximate cos(0.25) to 5 decimals of accuracy? Homework Equations taylor series...to complicated to type out here remainder of nth degree taylor polynomial = |R(x)| <= M/(n+1)! * |x - a|^(n+1)...

Homework Statement When a polynomial is divided by (x+2), the remainder is -19. When the same polynomial is divided by (x-1), the remainder is 2. Determine the remainder when the polynomial is divided by (x+2)(x-1). EDIT: Took out my attempts lol, there were way off. This was a "Math...

Homework Statement Prove that for any polynomial function f and number a, there exists a polynomial function g and number b such that: f(x) = (x-a)g(x) + b Homework Equations N/A The Attempt at a Solution Proof: Let P(n) be the statement that for some natural number n, f(x) =...

Homework Statement I understand How to do The remainder Theorem and The factor Theorem but I don't understand what they mean or what they are doing. I don't think I will be able to apply them without knowing what they mean. Can someone explain them to me? Homework Equations The...

Hi all, I had a problem, pls help me. Let b_1 < b_2 < \cdots < b_{\varphi(m)} be the integers between 1 and m that are relatively prime to m (including 1), of course, \varphi(m) is the number of integers between 1 and m that are relatively prime to m, and let B =...

1. The problem \statement, all variables and given/known data Estimate the error involved in using the first n terms for the function F(x) = \int_0^x e^{-t^2} dt Homework Equations The Attempt at a Solution I am using the Lagrange form of the remainder. I need to know the n+1 derivative of...

Homework Statement Find the remainder when (x^80 - 8x^30 + 9x^24 + 5x + 6) is divided by (x+1) Homework Equations The Attempt at a Solution So I'm not really sure where to start. I tried starting by doing long polynomial division, but I get stuck. How do I start this?

Homework Statement Find the remainder when 34! is divided by 37. Homework Equations Wilson's Theorem The Attempt at a Solution I understand that (p-1)! = (-1)(mod p) and that (p-2)! = (1)(mod p). I don't understand how to apply this to (p-3)! though.

Homework Statement Let f be a function whose seventh derivative is f7(x) = 10,000cos x. If x = 1 is in the interval of convergence of the power series for this function, then the Taylor polynomial of degree six centered at x = 0 will approximate f(1) with an error of not more than a.)...

Homework Statement Find the maximum error in approximating cos(x) by its Taylor polynomial of order 2 on the interval [ −.25, .25]. Justify your answer using the Remainder Estimation Theorem. Homework Equations |R3(x)<=M/3! |x|^3 The Attempt at a Solution |R3(x)<=M/3! |x|^3...

Homework Statement Prove that is m, n, and d are integers and d divides (m-n) then m mod d = n mod d. Homework Equations Quotient Remainder Theorem: Given any integer n and positive integer d, there exists unique integers q and r such that n=dq + r and 0\leqr<d and n mod d = r. The...

Chinese remainder theorem, urgent! Homework Statement This is an attempt to make the Chinese Remainder Theorem more concrete. Let m = 206 and n = 125. You may use the fact that 89n - 54m = 1. (a) What does the Chinese Remainder Theorem have to say about pairs of residues modulo 206 and...

Homework Statement I am trying to learn the Chinese Remainder Theorem from the following website: http://www.libraryofmath.com/chinese-remainder-theorem.html The only thing I don't understand is why the end result is expressed as another linear congruence. In the first example, the...

Hi, I can not see how this is implied... Let m and n be positive integers, with gcd(m, n) = 1. The the system of congruences x = a (mod m) and x = b (mod n ) has a solution. Moreover, any two solutions are congruent modulo mn. pf. Since gcd(m,n) = 1, there exist integers r...

This doesn't actually require the use of the CRT, since it actually wants you to sort of derive it for a system of two equations. So while using the CRT will help me solve this fairly quickly and easily, that's not what I'm after Homework Statement Let gcd(m,n)=1. Given integers a,b, show...

I know how the polynomial remainder theorem works but I can't see how knowing this is useful in any way. So I have f(X). I know that if I divide the statement in f(X) by X - a the remainder will be a. How is this useful knowledge though? What can I discover using this principle that I wouldn't...

Homework Statement The approximation e^{x}=1+x+(x^{2}/2) is used when X is small estimate the error when \left|x \right|<0.1Homework Equations \left|R_{n} \right|<\frac{M(x-a)^{n+1}}{(n+1)!}The Attempt at a Solution Since the Taylor expansion goes to the second power I used the third...

I stumbled upon this seamingle impossible question (without calculator!), any ideas to find the remainer of \frac{11^{345678}}{13}?

Find the lowest number that has a remainder of 1 when divided by 2, 2 when divided by 3, 3 when divided by 4, 4 when divided by 5, and 5 when divided by 6. It is possible to solve this by applying the general algorithm that solves Chinese Remainder problems. But, for this special...

Homework Statement In a 6 x 4 grid (6 rows, 4 columns), 12 of the 24 squares are to be shaded so that there are two shaded squares in each row and three shaded squares in each column. Let be the number of shadings with this property. Find the remainder when is divided by 1000. There is a...

Homework Statement Using the remainder theorem solve \frac{8x^{3} + 2x^{2} + 5}{2x^{2}+2} Homework Equations The remainder theorem: F(x) \equiv Q(x) \times divisor + remainder The Attempt at a Solution 8x^{3} + 2x^{2} + 5 \equiv (Ax+B)(2x^{2}+2) + C I can do it for...

I need help making sense of my notes: x congruent 4 mod 11 x congruent 3 mod 13 ai mi Mi yi aiMiyi 4 11 13 6 4*13*6 3 13 11 6 3*11*6 I'm not sure where the six came from

Homework Statement Find the 3rd-order Maclaurin Polynomial (i.e. P3,o(u)) for the function f(u) = sin u, together with an upper bound on the magnitude of the associated error (as a function of u), if this is to be used as an approximation to f on the interval [0,2]. I did the question...

Homework Statement When rx^3 + gx^2 +4x + 1 is divided by x-1, the remainder is 12. When it is divided by x+3, the remainder is -20. Find the values of r and g. Homework Equations The Attempt at a Solution r=f(1) =r(1)^3 + g(1)^2 + 4(1) +5 =r + g +9 r=12 r+g+9=12 r+g= 3...

A couple of hard questions about the Factor and Remainder Thoerem that I'm having a hard time with. Homework Statement 18) f(x) = 2x3 + x2 – 5x + c, where c is a constant. Given that f(1) = 0, (a) find the value of c (b) factorise f(x) completely, (c) find the remainder when...

Homework Statement How would you prove using number theory that C(n,m) is an integer where n => m =>1? Do you need the Chinese Remainder Theorem? It seems like it should follow easily from what C(n,m) represents but it is hard for me for some reason. Homework Equations The Attempt...

Homework Statement For all integers m, m^{}2=5k, or m^{}2=5k+1, or m^{}2=5k+4 for some integer k. Relevant equations I'm pretty sure we have to use the Quotient Remainder THM, which is: Given any integer n and positive integer d, there exists unique integers q and r such that...

Hi all! Could anyone help telling me the way to find the remainder of the following divisions: 1. (x^2006+x^1996+x^1981+x+1):(x^2-1) 2. (x2+x3+x5+1) : [(x-1)(x-2)] Thanks

I am doing a Chinese remainder theorem question and one of the equations is x \equiv 0 (mod 7). This would mean that x is a multiple of 7, but how do I use it in conjunction with the Chinese remainder theorem? Do I just ignore that equation, use the CRT on the rest of the system, and then once...