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Muh. Fadlan

a, b, c, d, dan e

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Diberikan sistem persamaan linear sebagai berikut.

$1 + a + b + c + d + e  = 1$

$32 + 16a + 8b + 4c + 2d + e = 2$

$243 + 81a + 27b + 9c + 3d + e = 3$

$1024 + 256a + 64b + 16c + 4d + e = 4$

$3125 + 625a + 125b + 25c + 5d + e = 5$ 

Tentukan nilai dari 

$(\frac{|a| + |b| + |c| + |d|}{|e|})^{2016}$

Edited by Muh. Fadlan
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Pandang polinom \(P(x)=x^5+ax^4+bx^3+cx^2+dx+e\)

Diketahui bahwa \(P(x)=x\) untuk \(x=1, 2, 3, 4, 5\)

Karena derajat tertinggi \(P\) adalah 5 dan \(P\) adalah polinom monik, maka

\(P(x)=(x-1)(x-2)(x-3)(x-4)(x-5)+x\)

dimana dari Vieta, kita tahu bahwa

\(|a|=1+2+3+4+5\)

\(|b|=1.2+1.3+1.4+1.5+2.3+2.4+2.5+3.4+3.5+4.5\)

\(|c|=1.2.3+1.2.4+1.2.5+1.3.4+1.3.5+1.4.5+2.3.4+2.3.5+2.4.5+3.4.5\)

\(|d|=1.2.3.4+1.2.3.5+1.2.4.5+1.3.4.5+2.3.4.5+1\)

dan

\(|e|=1.2.3.4.5\)

sehingga

\(\frac{|a|+|b|+|c|+|d|}{|e|}=\frac{|a|+|b|+|c|+|d|+|e|}{|e|}-1=\frac{(1+1).(1+2).(1+3).(1+4).(1+5)}{1.2.3.4.5}-1=6-1=5\)

Udah, tinggal dipangkat ama 2016, selesai

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