Remainder Definition and 171 Threads

I have a few questions about the remainder theorem. 1: For series that "skip" terms (example: 1+x^2+x^4+x^6) the theorem says the n+1 derivative and x^(n+1)/(n+1)!. For example if you have 1 + x^2 where you know the next term would be x^4 you could treat it as a third order or a...

How is the Taylor remainder of a series (with given Taylor expansion) expressed if you want to make a calculation with known error? e.g. if I want to calculate π to, say, 12 decimal places using the previously-derived result π=4*arctan(1) and the Taylor series for arctan(x), how will I work out...

I'm tutoring a pupil for a CLEP exam and her book includes the following algebra problem: What is the remainder when 9x^{23} - 7x^{12} - 2x^{5} +1 is divided by x+1 ? I know how to find the answer by computing the quotient of these two expressions, but in this case doing that is so tedious I...

Homework Statement The goal of this problem is to approximate the value of ln 2. We will use two different approaches: (a) First, we use the Taylor polynomial pn(x) of the function f(x) = lnx centered at a = 1. Write the general expression for the nth Taylor polynomial pn(x) for f(x) = lnx...

Homework Statement Q. How to find Remainder of the following expression ##32^{32^{32}}\mod{}6=?## Homework Equations The Attempt at a Solution ##32^{32}\mod{}2=0 \implies 32^{32}=2x## ##32^{32}\mod{}3=1 \implies 32^{32}=3y+1## would it help to find the remainder of ##32^{32^{32}}\mod{}6##

Isn't it amusing ?What could be the probable explanation for this?Also when operated by division operator gives the rest of the number as the quotient (Note only when the divisor is 10)

please find the remainder $\dfrac {19^{81}+19^{49}+19^{25}+19^9+19}{19^3-19}$

Homework Statement Theorm: Let m and n be relatively prime integers. If s and t are arbitrary integers there exists a solution x in Z to the simultaneous congruences: x~s (mod m) and x~t (mod n) Part of proof that confuses me: Since gcd(m,n) = 1, the Euclidean algorithm gives p and q in...

find the remainder when $3^{2^n}-1$ is divided by $2^{n+3}$

Homework Statement At what points ##x## in the interval ##(-1,1]## can one use the Lagrange Remainder Theorem to verify the expansion ##ln(1+x)=\sum_{k=1}^{\infty} (-1)^{k+1}{\frac{x^k}{k!}}##Homework Equations The Attempt at a Solution Now I know that ##ln(1+x)=\sum_{k=1}^{\infty}...

Problem: When $3^{2002}+7^{2002}+2002$ is divided by 29, the remainder is: A)0 B)1 C)2 D)7 Attempt: I tried the following: $3^3 \equiv -2\mod 29$ i.e $3^{2002} \equiv 3\cdot 3^{2001} \equiv -3\cdot 2^{667} \mod 29$ Also, $2^5 \equiv 3 \mod 29$, hence, $-3\cdot 2^{667} \equiv -3\cdot...

Homework Statement Prove that ##(a-b)## is a factor of ##a^5-b^5##, and find the other factor. Homework Equations Remainder theorem : remainder polynomial ##p(x)## divided by ##(x-a)## is equal to ##p(a)## Factor theorem : if remainder = 0, then divisor was a factor of dividend...

Q2.) Show all working out. a) Find the remainder when $$x^3+2x^2-5x-3$$ is divided by $$x-2$$. b) Find the remainder when $$x^3-3x^2-x+3$$ is divided by $$x-3$$.

Q1.) Use the factor and remainder theorems to find solutions to: $$1x^3+1x^2+-9x+D=0$$

Homework Statement Let Tn(x) be the degree n polynomial of the function sin x at a=0. Suppose you approx f(x) by Tn(x) if abs(x)<=1, how many terms are need (what is n) to obtain an error less than 1/120 Homework Equations Rn(x)=M(x-a)^(n+1)/(n+1)! sin(x)=sum from 0 to ∞ of...

Homework Statement Find each remainder: a. (x^3 + 5x^2 - 7x + 1) ÷ (x+2)(x-1)b. (2x^3 + x^2 - 4x - 2) ÷ (x^2 + 4x + 3)Homework Equations N/A. (We've used Long Division and Synthetic Division for previous questions.) The Attempt at a Solution How would i go about solving these? I'm pretty stuck.

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-1)(x+2)Homework Equations The Attempt at a Solution had the polynomial been a real...

What is the most tangible way to introduce the Chinese Remainder Theorem? What are the practical and really interesting examples of this theorem. I am looking for examples which have a real impact on students.

Chinese Remainder Theorem What is the most tangible way to introduce the Chinese Remainder Theorem? What are the practical and really interesting examples of this theorem. I am looking for examples which have a real impact on students.

Find the remainder when 1992 is divided by 92

Homework Statement (-8)^4124 + 6^3101 + 7^5 is divided by 3. Homework Equations The Attempt at a Solution My original insight was that 6 raised to any power is always divisible by 3. 7 raised to any power yields a remainder of 1 when divided by 3. and the remainder from -8...

Consider a sequence given by $$a_n=a_{n-1}+3a_{n-2}+a_{n-3}$$, where $$a_0=a_1=a_2=1$$. What is the remainder of $$a_{2013}$$ divided by $$7$$?

Consider the triangular array of numbers $$0,\;1,\;2,\;3,\cdots$$ along the sides and interior numbers obtained by adding the two adjacent numbers in the previoius row. Row 1 through row 6 are shown as below. 0 1 1 2 2 2 3 4 4 3 4 7 8 7 4 5...

Homework Statement This is not a homework problem but I would like to clarify my concern. It is stated that a function can be written as such: f(x) = \lim_{n \rightarrow ∞} \sum^{∞}_{k=0} f^{(k)} \frac{(x-x_{0})^k}{k!} R_{n}=\int^{x}_{x_{0}} f^{(n+1)} (t) \frac{(x-t)^n}{n!} dt They...

Homework Statement If a , b, c are distinct and p(x) is a polynomial in x which leaves remainders a,b,c on division by (x-a),(x-b),(x-c) respectively. Then the remainder on division of p(x) by(x-a)(x-b)(x-c) is Homework Equations As it is given that p(x) gives remainder a when divided by...

Homework Statement Find all integers x such that 7x \equiv 11 mod 30 and 9x \equiv 17 mod 25 Homework Equations I guess the Chinese Remainder theorem and Bezout's theorem would be used here. The Attempt at a Solution I can do this if the x-terms didn't have a...

Is there a generalization for the Chinese Remainder Theorem if the modular bases are not coprime? Or even to some extent, if the modular bases are increasing by the same common ratio? I searched it up but could not find anything.

Homework Statement Consider the followign function f(x) = x^-5 a=1 n=2 0.8 \leq x \leq 1.2 a) Approximate f with a tayloy polynomial of nth degree at the number a = 1 b) use taylor's inequality to estimate the accuracy of approximation f(x) ≈ T_{n}(x) when x lies in the interval...

Hi, Homework Statement I am trying to limit Lagrange's remainder on taylor expansion of ln(4/5) to be ≤ 1/1000. Homework Equations The Attempt at a Solution I have tried using both ln(1+x), where x=-1/5 and x0(the center)=0, and ln(x), where x=4/5 and x0=1. Every time I keep...

Homework Statement The question asks to estimate the remainder on the interval |x|≤ 1. f(x) is given as sinh(x). I solved the polynomial P3(x) = x + (1/6)(x3) I then went ahead and solved R3(x) up to the point shown below. R3(x) = (sinh(c)*x4)(1/24)I then don't know how to go about...

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...

Homework Statement Find the remainder of \frac{25!}{29} Homework Equations The Attempt at a Solution One of my friend asked me this question and i was clueless how should i start? (I am not sure that the question is correct.) Any help is appreciated!

If I was to try to work this out I would use the Chinese Remainder Theorem and since 10557 = 3^3 . 17 . 23 end up with (Z/10557Z)* isomorphic to (Z/27Z)* x (Z/17Z)* x (Z/23Z)* isomorphic to C18 x C16 x C22 where Cn represents the Cyclic group order n. How would I then write this as Cn1 x Cn2...

Homework Statement A) Find the remainder of 2^n and 3^n when divided by 5. B)Conclude the remainder of 2792^217 when divided by 5. C)solve in N the following : 1) 7^n+1 Ξ 0(mod5) 2) 2^n+3^n Ξ 0(mod5) The Attempt at a SolutionA) I know that for the first two I have to get 2^n=5k+r and...

1. Prove that the MacLaurin series for cosx converges to cosx for all x. Homework Equations Ʃ(n=0 to infinity) ((-1)^n)(x^2n)/((2n)!) is the MacLaurin series for cosx |Rn(x)|\leqM*(|x|^(n+1))/((n+1)!) if |f^(n+1)(x)|\leqM lim(n->infinity)Rn=0 then a function is equal to its Taylor series...

I have to prove this problem. For all n integers, if n mod 5 = 3, then n2 mod 5 = 4 Proof: Let n be an integer such that n mod 5 = 3. n = 5k+3 for some integer k by definition of MOD or QRT? Which one would be correct? Am I using the definition of MOD or QRT? I'm thinking its QRT because its...

It asks "use remainder estimate for integral test" to find series accurate to 3 dec? Homework Statement It says "Use the Remainder Estimate for the Integral Test to find the sum of the following series to three decimal places of accuracy." \sum^{\infty}_{n=1} \frac{1}{n^{3}} Homework...

Homework Statement when f(x) is divided by (x+1), remainder is -9; when f(x) is divided by (x-3), remainder is -1; what is the remainder if f(x) is divided by (x+1)(x-3)? Homework Equations f(x) = divider * q(x) + remainder The Attempt at a Solution f(x) = (x+1) * a(x) -9 f(x) =...

Homework Statement What is the minimal degree Taylor polynomial about x=0 that you need to calculate sin(1) to 3 decimal places? 6 decimal places? Homework Equations R_nx = f^(n+1)(c)(x-a)^(n+1)/(n+1)(factorial) The Attempt at a Solution I have attached my attempt. I am stuck on the...

Let's say I'm trying to find the oblique asymptote of the function: f(x)= -3x2 + 2 x-1 Forgive my poor formatting. So because the denominator isn't linear, we do polynomial long division of the function and ultimately get -3x - 3 as our quotient, with a remainder of...

So we had two examples in class, but I don't understand why they're different. And the professor is away today, which means I won't see him until the entire weekend has passed (a nightmare for students like me who obsess over a problem). 1. For which x is the approximation sin(x) ≈ x - (x^3)/6...

Homework Statement Find the Maclaurin series of f(x) = x^2cos(x) Homework Equations I got the answer to be (sum from n=1 to infinity) \frac{(-1)(^n+1)x(^2n)}{(2n-2)!} and the formula for the remainder is R_n(x) = \frac{f(^n+1)(c)}{(n+1)!}x(^n+1) (I have no idea how to make those exponents...

Homework Statement i = the 3x3 matrix below 2-λ 0 1 -1 4-λ -1 -1 2 0-λ Using remainder and factor theorem find the 3 values of λ. Homework Equations |i| = a1|b2c3-c2b3|-a2|a2c3-c2a3|+a3|a2b3-b2a3| |a|=ad-bc The Attempt at a Solution(2-λ) |(4-λ x...

Suppose I have two series A=\sum_{n=0}^\infty a_n B=\sum_{n=0}^\infty b_n and I have estimates for the remainders of each one: \sum_{n=N}^\infty a_n \le R^N_A \sum_{n=N}^\infty b_n \le R^N_B Consider the product series AB=\sum_{n=0}^\infty c_n where c_n=\sum_{i=0}^n a_i...

I was self-studying Discrete Mathematics and I found the following problem. Homework Statement Show that whenever a and b are positive integers, then (2^a-1)\mod(2^b-1)=2^{a\mod b}-1 The attempt at a solution I don't know if there is a more rigorous way of solving it, but I came up with the...

Homework Statement Prove that for every pair of numbers x and h, \left|sin\left(x+h\right)-\left(sinx+hcosx\right)\right|\leq\frac{h^{2}}{2} The Attempt at a Solution Let f(x)= \left|sin\left(x+h\right)-\left(sinx-hcosx\right)\right|? and then to center the taylor polynomial around 0...

Homework Statement Hi! This is not a homework question but it came in my test. What is the remainder when 4101 is divided by 101? Homework Equations (1+x)n-1 is always divisible by x. The Attempt at a Solution Only the equation given above came in my mind but i wasn't able to...

Hi, I am looking for an explanation, if any, on why every integer square leaves remainder 0 or 1 on division by 4. Appreciate your time and help bluemoon2188

Usually to do the remainder we take Rn(x) = (f differentiated n+1 times at a ).(x-c)n+1/(n+1)!, but when my function is sin(x) do i take (f differentiated 2n+2 times at a ).(x-c)2n+2/(2n+2)!? Thanks

Homework Statement struggling to understand how to solve this question and would be extrememly grateful if someone who understood offered me a hand: need to find the remainder of gogool divided by 1001. i.e 10^100mod1001. any assistance would be much welcomed as i have no hope figuring...