too large expression... :O

Algebra Level 3

If one root of the equation x⁴ − 9x³ + 4x² + 234x − 680 = 0 is 5 + 3i. Find the sum of all roots.


The answer is 9.

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Christian Daang by Christian Daang , 20, Philippines

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7 solutions

Jun Arro Estrella
Nov 15, 2014

hahahah.. can be done by inspection.. the sum is on the degree n-1 so it follows that -(-9)=9

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Indulal Gopal
Nov 15, 2014

A simple application of Vieta's formula which is classical in the theory of equations readily gives the answer as 9.The information regarding one root is only a joke from the poser to mislead solvers :)

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Joel Tan
Nov 15, 2014

By Vieta's formula sum of roots is -(-9)=9.

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Sorry just saw @Calvin Lin 's post

Joel Tan - 6 years, 7 months ago
Raushan Sharma
Jul 7, 2015

Every information given other than the equation is just to befool us. We can simply get the sum of roots as the negative of the coefficient of x 3 x^3 that is -(-9) = 9

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Ramiel To-ong
May 29, 2015

Using Vietas formula

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Just apply Vieta's formula and you get the answer as 9 \boxed{9}

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Christian Daang
Nov 8, 2014

Answer: 9

If the 1st root is 5 + 3i,

then,

x = 5+3i

x-5 = 3i , ==>square both sides

x^2 - 10x + 25 = -9

x^2 - 10x + 34 = 0

x⁴ − 9x³ + 4x² + 234x − 680 = 0 -----> (x²-10x+34)(x²+yx-20) = 0 for some y.

(x²-10x+34)(x²+yx-20) = 0 -----> x⁴-x³(10-y)+x²(14-10y)+x(200+34y)-680 = 0 = x⁴ − 9x³ + 4x² + 234x − 680 .

By comparing the 2 equations,. we can see that y = 1.then,

x⁴ − 9x³ + 4x² + 234x − 680 = 0 -----> (x²-10x+34)(x²+yx-20) = 0 --------> (x²-10x+34)(x²+x-20) = 0 --------> (x²-10x+34)(x+5)(x-4) = 0

therefore,

the other 3 roots are: {5-3i , -5 , +4} .

Adding all of them, 10+4-5 = 9..

Final Answer: 9

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Please check that you are receiving emails regarding your dispute. The submitted disputes state "the sum of roots is 9 , the roots are -5 , 4, 5 - 3i , 5 + 3i"

Note that by Vieta's formula, the sum of roots is just − a 3 a 4 = − − 9 1 = 9 - \frac{ a_3} { a_4} = - \frac{-9}{1 } = 9 . You made some calculation mistakes above, specifically in factoring x 2 + x − 20 x^2 + x - 20 .

Calvin Lin Staff - 6 years, 7 months ago

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ya,.. The factor should be: (x+5)(x-4) and not, (x-5)(x+4).

Christian Daang - 6 years, 7 months ago

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