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Algebra Level 4

If x = 3 + 3 1 / 3 + 3 2 / 3 x = 3 + 3^{1/3} + 3^{2/3} , find x 3 − 9 x 2 + 18 x − 10 x^3 - 9x^2 + 18x - 10 .

3 4 7 5 1 6 0 2

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Soumya Shrivastva by Soumya Shrivastva , 20, India

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2 solutions

Chew-Seong Cheong
Jan 16, 2016

Let a = 3 1 3 a = 3^{\frac{1}{3}} , then x = 3 + a + a 2 x = 3+ a + a^2 .

x 3 − 9 x 2 + 18 x − 10 = x 3 − 9 x 2 + 27 x − 27 − 9 x + 17 = ( x − 3 ) 3 − 9 x + 17 = a 3 ( 1 + a ) 3 − 9 ( 3 + a + a 2 ) + 17 = 3 ( 1 + 3 a + 3 a 2 + 3 ) − 27 − 9 a − 9 a 2 + 17 = 12 − 27 + 17 = 2 \begin{aligned} x^3 - 9x^2 + 18x - 10 & = x^3 - 9x^2 + 27x - 27 - 9x + 17 \\ & = (x-3)^3 - 9x + 17 \\ & = a^3(1+a)^3 - 9(3 + a + a^2) + 17 \\ & = 3(1+3a+3a^2 +3) - 27 - 9 a - 9a^2 +17 \\ & = 12-27+17=\boxed{2}\end{aligned}

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Mamunur Rashid
Jan 15, 2016

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