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Algebra Level 5

Let a , b a, b and c c be the distinct roots of the polynomial P ( x ) = x 3 − 􀀀 10 x 2 + x 􀀀 − 2015 P(x) = x^3-􀀀10x^2 +x􀀀-2015 . If the cubic polynomial Q ( x ) Q(x) is monic and has distinct roots b c − 􀀀 a 2 , c a − 􀀀 b 2 bc-􀀀a^2, ca-􀀀b^2 and a b − c 2 ab-c^2 , what is the sum of the coefficients of Q ( x ) Q(x) ?


The answer is 2015000.

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Department 8 by Department 8 , 27, Israel

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1 solution

Department 8
Mar 14, 2016

Considering the factorization of Q, ( x − b c + a 2 ) ( x − c a + b 2 ) ( x − a b + c 2 ) (x-bc+a^2)(x-ca+b^2)(x-ab+c^2) , since we are asked the sum of coefficients we have plug x = 1 x=1 .

So we have to find ( 1 − 􀀀 b c + a 2 ) ( 1 − 􀀀 c a + b 2 ) ( 1 − 􀀀 a b + c 2 ) (1-􀀀bc+a^2)(1-􀀀ca+b^2)(1-􀀀ab+c^2) . Since 1 = a b + b c + c a 1 = ab + bc + ca by Vieta's Formulas, this rewrites as:

( a ( a + b + c ) ) ( b ( a + b + c ) ) ( c ( a + b + c ) ) = a b c ( a + b + c ) 3 = 2015000 \large{(a(a + b + c))(b(a + b + c))(c(a + b + c)) = abc(a + b + c)^3 = 2015000}

4 Helpful 0 Interesting 1 Brilliant 0 Confused

That's a pretty great method :-)

Pulkit Gupta - 5 years, 3 months ago

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Thank you:-)

Department 8 - 5 years, 3 months ago

Same but only before your nice trick!! I expanded it and then used Vieta's. Happy to be second solver.

Aakash Khandelwal - 5 years, 3 months ago

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Thanks, because of you expanded I want to tell my friends did the same as yours so that's why I have this nice title

Department 8 - 5 years, 3 months ago

Same approach :) Thumbs up for that.

Jun Arro Estrella - 4 years, 5 months ago

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