Ahaan's Roots

Algebra Level 4

If the roots of the polynomial f ( x ) = x 3 − x + 4 f(x)=x^3-x+4 are r r , s s , and t t , find ∣ r 2 s + r 2 t + r s 2 + r t 2 + s 2 t + s t 2 ∣ . |r^2s+r^2t+rs^2+rt^2+s^2t+st^2| .

This problem is proposed by Ahaan Rungta .


The answer is 12.

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Calvin Lin by Calvin Lin

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1 solution

By Vieta's we have r + s + t = 0, rs + rt + st = -1 and rst = -4. Let the expression inside the absolute value signs be S.

Then (r + s + t) * (rs + rt + st) = S + 3 * rst = 0, and so S = -3 * rst = 12.

So finally l S l = 12.

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Actually, r s t = ( − 4 ) rst=(-4) by Vieta. So, we have S = 12 S=12 . The absolute value signs in the question doesn't change the answer.

I think you should edit your answer.

Prasun Biswas - 5 years, 11 months ago

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Done. Thanks for catching the mistake. :)

Brian Charlesworth - 5 years, 11 months ago

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