Problem 2

Algebra Level 4

Let p , q p,q and r r be the roots of equation x 3 + 3 x + 3 = 0 x^3 + 3x + 3 = 0 . Find the value of p 5 + q 5 + r 5 p^5+q^5+r^5 .


The answer is 45.

Ratul Pan by Ratul Pan , 21, India

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1 solution

Akshat Sharda
May 19, 2016

x 3 + 3 x + 3 = 0 x 3 = − 3 x − 3 x 5 = − 3 x 3 − 3 x 2 ∑ x 5 = − 3 ∑ x 3 − 3 ∑ x 2 \begin{aligned} x^3+3x+3 &=0 \\ x^3 &= -3x-3 \\ x^5 &=-3x^3-3x^2 \\ \sum x^5 &=-3 \sum x^3-3\sum x^2 \end{aligned}

By Newton's sum ∑ x 2 = − 6 \sum x^2=-6 and ∑ x 3 = − 9 \sum x^3=-9 .

Therefore, ∑ x 5 = 18 + 27 = 45 \sum x^5=18+27=\boxed{45}

1 Helpful 0 Interesting 0 Brilliant 0 Confused

But its only real root is around -0.8 and the other two should be conjugate pairs of each other. Then how can the answer be positive??

Prithwish Roy - 4 years, 3 months ago

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