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• Polynomial Remainder Theorem
  ...tic division | dividing]] any [[polynomial]] <math>P(x)</math> by a linear polynomial <math>x-a</math>, both with [[Complex number|complex]] coefficients, is equ ...and divisor <math>x-a</math>, that exist a quotient <math>Q(x)</math> and remainder <math>R(x)</math> such that <cmath>P(x) = (x-a) Q(x) + R(x)</cmath> with <m
  3 KB (415 words) - 13:00, 11 March 2025
• Proof of the Polynomial Remainder Theorem
  ...g from Algebra to Number Theory. This depicts how important the polynomial remainder theorem truly is, and why it must be taught in all courses and is a great t ...>\frac{f(x)}{x-a}=f(a)</math> Written below is the proof of the polynomial remainder theorem.
  2 KB (298 words) - 18:15, 30 April 2019
• Polynomial remainder theorem
  #REDIRECT[[Polynomial Remainder Theorem]]
  41 bytes (4 words) - 16:42, 27 February 2022

Page text matches

• Wilson's Theorem
  ...s Little Theorem]], every nonzero element of this field is a root of the [[polynomial]] ...\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{23}=\frac{a}{23!}</math>. Find the remainder when <math>a</math> is divided by <math>13</math>.
  4 KB (639 words) - 01:53, 2 February 2023
• Algebra/Intermediate
  ** [[Remainder theorem]] ** [[Elementary symmetric polynomial]]
  2 KB (198 words) - 16:06, 7 December 2024
• Modular arithmetic/Intermediate
  ...ave a number of roots greater than its degree. Consider, for example, the polynomial congruence ...four solutions -- two more than we might expect based on the degree of the polynomial.
  14 KB (2,317 words) - 19:01, 29 October 2021
• 2003 AMC 12A Problems
  ...number, and let <math>q</math> and <math>r</math> be the quotient and the remainder, respectively, when <math>n</math> is divided by <math>100</math>. For how The graph of the polynomial
  13 KB (1,955 words) - 21:06, 19 August 2023
• 1988 AIME Problems/Problem 13
  ...on|recursive definition, <math>F_{n+2} - F_{n+1} - F_n = 0</math>, and the polynomial <math>x^2 - x - 1 = 0</math>. ...follow the Fibonacci sequence. Carrying out this pattern, we find that the remainder is <cmath>(F_{16}b + F_{17}a)x + F_{15}b + F_{16}a + 1 = 0.</cmath> Since t
  10 KB (1,595 words) - 16:30, 24 August 2024
• 2007 iTest Problems
  Find the remainder when <math>N</math> is divided by <math>25</math>. Find the remainder when Ted's favorite number is divided by 25.
  30 KB (4,794 words) - 23:00, 8 May 2024
• Mock AIME 6 Pre 2005/Problems
  <math>1</math>. Find the remainder when <math>\displaystyle 11^{2005}</math> is divided by <math>\displaystyle ...05\sqrt{\cdots}}}</math> where <math>\displaystyle f(x)>0</math>. Find the remainder when <math>\displaystyle f(0)f(2006)f(4014)f(6024)f(8036)</math> is divided
  7 KB (1,110 words) - 05:15, 31 December 2006
• Mock AIME 4 2006-2007 Problems
  ...ath>r_1, r_2, \ldots , r_{2006}</math> are distinct integers such that the polynomial <math>(x-r_{1})(x-r_{2})\cdots (x-r_{2006})</math> has exactly <math>n</mat Find the remainder when <math>3^{3^{3^3}}</math> is divided by 1000.
  6 KB (923 words) - 14:17, 16 January 2007
• Asymptote (geometry)
  ...of <math>\frac{P(x)}{Q(x)}</math> where <math>P(x), Q(x)</math> are both [[polynomial]]s: 1. If the [[Degree of a polynomial | degree ]] of <math>Q(x)</math> is greater than that of the degree of <mat
  4 KB (666 words) - 11:46, 28 September 2024
• Nonconstant
  ...(if there is more than one element in its [[range]]). For example, the [[polynomial]] <math>p(x) = x^2 - x + 1</math> with the [[real number]]s as [[domain]] a ...e value of <math>x^5 -2x^4 -2x^3 - x^2 + x + 4</math> and then takes the [[remainder]] of this number on division by 3 appears quite complicated but turns out t
  1 KB (192 words) - 11:00, 10 May 2008
• Mock AIME 2 Pre 2005/Problems
  ...rds can be dealt to each player such that this is the case. Determine the remainder obtained when <math>N</math> is divided by <math>1000</math>. (Bridge is a Determine the remainder obtained when the expression
  6 KB (990 words) - 15:23, 11 November 2009
• 2007 AIME I Problems
  Find the remainder when <math>N</math> is divided by 1000. (<math>\lfloor{k}\rfloor</math> is The polynomial <math>P(x)</math> is cubic. What is the largest value of <math>k</math> fo
  7 KB (1,218 words) - 15:28, 11 July 2022
• 2007 AIME II Problems
  ...be the maximum possible number of basic rectangles determined. Find the [[remainder]] when <math>N</math> is divided by 1000. Let <math>f(x)</math> be a [[polynomial]] with real [[coefficient]]s such that <math>f(0) = 1,</math> <math>f(2)+f(
  9 KB (1,435 words) - 01:45, 6 December 2021
• 2007 USA TST Problems
  ...of <math>\{1, 3, 5, ..., 1023\}</math>. Show that the number of complete remainder sequences is at most <math>2^{35}</math>.
  3 KB (519 words) - 20:59, 24 May 2007
• 1999 AHSME Problems
  ...ber of points of intersection of the graphs of two different fourth degree polynomial functions <math>y = p(x)</math> and <math>y = q(x)</math>, each with leadin ...ided by <math>x - 99</math>, the remainder is <math>19</math>. What is the remainder when <math>P(x)</math> is divided by <math>(x-19)(x-99)</math>?
  13 KB (1,945 words) - 21:04, 2 February 2025
• Factor Theorem
  ...eorem''' is a theorem regarding the relationships between the factors of a polynomial and its roots. ...</math> is constant, <math>f</math> is polynomial) is <math>0</math> using polynomial division rather than plugging in large values.
  3 KB (510 words) - 18:48, 25 February 2025
• Mock AIME 1 2007-2008 Problems
  .../math> and <math>n</math> are relatively prime positive integers, find the remainder when <math>m+n</math> is divided by <math>2008</math>. Let <math>F(x)</math> be a polynomial such that <math>F(6) = 15</math> and
  6 KB (992 words) - 14:15, 13 February 2018
• 2008 USAMO Problems/Problem 1
  .../math> be respective residues as described in the lemma. By the [[Chinese Remainder Theorem]], there is a positive integer <math>a</math> that satisfies the re ...math>2 < p_0 < p_1 < \cdots < p_n</math> be such primes. Using the Chinese Remainder Theorem to specify <math>a</math> modulo <math>p_n</math>, we can find an i
  11 KB (1,964 words) - 03:38, 17 August 2019
• 2008 iTest Problems
  ...n, but none receive the same number of coins, then Tony and Nick split the remainder equally. If there are <math>2000</math> gold coins in the chest, what is th ...math>2008</math>, including <math>1</math> and <math>2008</math>. Find the remainder when <math>S</math> is divided by <math>1000</math>.
  71 KB (11,749 words) - 12:39, 20 November 2024
• 2008 iTest Problems/Problem 80
  and let <math>r(x)</math> be the polynomial remainder when <math>p(x)</math> is divided by <math>x^4+x^3+2x^2+x+1</math>. Find the remainder when <math>|r(2008)|</math> is divided by <math>1000</math>.
  3 KB (560 words) - 19:49, 23 November 2018

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