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  • An '''elementary symmetric sum''' is a type of [[summation]]. The <math>k</math>-th '''elementary symmetric sum''' of a [[set]] of <math>n</math> numbers is the sum of all products of
    2 KB (275 words) - 11:51, 26 July 2023
  • ** [[Elementary symmetric polynomial]]
    2 KB (198 words) - 15:06, 7 December 2024
  • ...give us a clever and efficient way of finding the sums of [[root]]s of a [[polynomial]] raised to a power. They can also be used to derive several [[factoring]] Consider a polynomial <math>P(x)</math> of degree <math>n</math>,
    4 KB (704 words) - 07:28, 24 November 2024
  • The '''fundamental theorem of algebra''' states that every [[nonconstant]] [[polynomial]] with [[complex number|complex]] [[coefficient]]s has a complex [[root]]. ...th>n</math> complex roots, counting multiplicities. In other words, every polynomial over <math>\mathbb{C}</math> splits over <math>\mathbb{C}</math>, or decomp
    5 KB (832 words) - 13:22, 11 January 2024
  • A set <math>S</math> of points in the <math>xy</math>-plane is symmetric about the origin, both coordinate axes, and the line <math>y=x</math>. If < The graph of the polynomial
    13 KB (1,955 words) - 20:06, 19 August 2023
  • ...c against line <math>x=\frac{a}{2}</math>. Try this yourself by graphing a polynomial <math>f(x)</math>, then graphing <math>f(n-x)</math>. If <math>f(x)=f(n-x)< ...e lines <math>x=1079</math> and <math>x=1607</math> due to it being, well, symmetric. Doing so, we find that there will be two new lines of symmetry generated b
    5 KB (921 words) - 15:36, 28 November 2024
  • ...lambda}{4} + \frac{1}{4}</math>, and extracting eigenvalues by setting the polynomial equal to <math>0</math>, we have 2 eigenvalues: <math>\lambda_1 = 1</math> ...math>A</math>. Now for <math>B</math> and <math>C</math>. Oh wait they are symmetric. So then if this is the correct answer, why am I wrong, or what happened to
    15 KB (2,406 words) - 22:56, 23 November 2023
  • ...>2\sqrt{10}(t^3 + 3t) = 200x^3 - \frac{2}{10x^3}</math>, which reduces the polynomial to just <math>(t^2 + 3)\left(2\sqrt{10}t + 1\right) = 0</math>. Then one ca It's symmetric! Dividing by <math>y^3</math> and rearranging, we get
    7 KB (1,098 words) - 23:33, 20 January 2025
  • ..._3 + \zeta_3\zeta_1,\ e_3 = \zeta_1\zeta_2\zeta_3</math> (the [[elementary symmetric sums]]). Then, we can rewrite the above equations as ...ath>\zeta_1, \zeta_2,</math> and <math>\zeta_3</math> are the roots of the polynomial
    2 KB (221 words) - 01:49, 19 March 2015
  • ...h> \prod_{i=1}^{n}(t+x_i) </math> (see [[Viete's sums]]). We define the ''symmetric average'' <math>d_k </math> to be <math> \textstyle s_k/{n \choose k} </mat ...ath>, there exist real <math> x'_1, \ldots, x'_{m-1} </math> with the same symmetric averages <math> d_0, \ldots, d_{m-1} </math>.
    5 KB (830 words) - 22:30, 13 January 2025
  • '''Maclaurin's Inequality''' is an inequality in [[symmetric polynomial]]s. For notation and background, we refer to [[Newton's Inequality]]. * [[Symmetric sum]]
    992 bytes (146 words) - 15:48, 29 December 2021
  • The '''symmetric sum''' <math>\sum_{\rm sym} f(x_1, x_2, x_3, \dots, x_n)</math> of a funct More generally, a '''symmetric sum''' of <math>n</math> variables is a sum that is unchanged by any [[perm
    1 KB (255 words) - 11:52, 8 October 2023
  • ...mials]] of its roots can be easily expressed as a ratio between two of the polynomial's coefficients. It is among the most ubiquitous results to circumvent finding a polynomial's roots in competition math and sees widespread usage in many math contests
    3 KB (528 words) - 20:24, 21 October 2024
  • ...h>, is the group of permutations on <math>M</math>. A [[subgroup]] of the symmetric group on <math>M</math> is sometimes called a '''permutation group''' on <m ...<math>n</math> is <math>S_{n}</math>, and this can be used to prove that [[polynomial]] [[equation]]s of degree five and higher are unsolvable through the use of
    10 KB (1,668 words) - 14:33, 25 May 2008
  • ...us <math>x</math>, <math>y</math>, and <math>z</math> are the roots of the polynomial <math>t^3-3t^2+3t-1=(t-1)^3</math>. Thus <math>x=y=z=1</math>, and there ar So now we have three equations for the elementary symmetric sums of <math>x,y,z</math>:
    5 KB (888 words) - 07:18, 22 April 2024
  • The polynomial <math>p</math> is symmetric. Each symmetric polynomial has the following property: for <math>x\not=0</math>, <math>x</math> is a r Our polynomial has six complex roots: <math>\alpha</math>, <math>1/\alpha</math>, <math>\b
    4 KB (615 words) - 07:09, 26 January 2009
  • The '''characteristic polynomial''' of a linear [[operator]] refers to the [[polynomial]] whose roots are the [[eigenvalue]]s of the operator. It carries much info In the context of problem-solving, the characteristic polynomial is often used to find closed forms for the solutions of [[#Linear recurrenc
    19 KB (3,412 words) - 13:57, 21 September 2022
  • ...the equation <math> f(x)=0 </math> has exactly four distinct real [[Root (polynomial)|roots]], then the sum of these roots is ...the other two roots will be less than <math>2</math>. These four roots are symmetric around <math>2</math>, so the average of the four roots is <math>2</math>.
    1 KB (242 words) - 12:53, 24 January 2024
  • ...> -12k^2+72k-12\geq0 </math>, or <math> k^2-6k+1\leq0 </math>. The [[Root (polynomial)|roots]] of this quadratic, using the [[quadratic formula]], are <math> 3\p Since <math>x</math> and <math>y</math> are symmetric, we want <math>x</math> to be the lower number to maximize <math>k</math>,
    4 KB (614 words) - 19:09, 12 September 2022
  • Since <math>S</math> is a [[Elementary symmetric sum|symmetric polynomial]] of degree <math>2</math>, we try squaring the first equation to get:
    1 KB (220 words) - 15:24, 28 June 2021

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