PUMac 2015 Algebra A Problem 3

Algebra Level 4

Find the sum of the non-repeated roots of the polynomial P ( x ) = x 6 − 5 x 5 − 4 x 4 − 5 x 3 + 8 x 2 + 7 x + 7 P(x) = x^6-5x^5-4x^4-5x^3+8x^2+7x+7 .


The answer is 7.

Alan Yan by Alan Yan , 20, USA

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2 solutions

Alan Yan
Nov 22, 2015

P ( x ) = x 6 − 5 x 5 − 4 x 4 − 5 x 3 + 8 x 2 + 7 x + 7 = ( x 6 − 5 x 5 − 4 x 4 − 5 x 3 + x 2 ) + 7 ( x 2 + x + 1 ) = ( x 2 + x + 1 ) ( x 4 − 6 x 3 + x 2 + 7 ) = ( x 2 + x + 1 ) 2 ( x 2 − 7 x + 7 ) \begin{aligned} P(x) & = x^6-5x^5-4x^4-5x^3+8x^2+7x+7 = (x^6 - 5x^5 - 4x^4 -5x^3 + x^2)+7(x^2+x+1) \\ & = (x^2+x+1)(x^4 - 6x^3 + x^2 + 7) = (x^2+x+1)^2(x^2 - 7x + 7) \end{aligned} So the answer is 7 \boxed{7} .

2 Helpful 0 Interesting 0 Brilliant 0 Confused
Otto Bretscher
Nov 22, 2015

We observe that the non-trivial third roots of unity, ω \omega and ω 2 \omega^2 , are double roots of P ( x ) P(x) as they are roots of both P ( x ) P(x) and P ′ ( x ) P'(x) . By Viete, the sum we seek is 5 − 4 ( − 1 2 ) = 7 5-4\left(-\frac{1}{2}\right)=\boxed{7} .

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