Polynomial Sublimation

Algebra Level 3

Let α , β , θ \alpha, \beta, \theta be the roots of the cubic equation x 3 − 12 x 2 + 39 x − k = 0 x^3 -12x^2 +39x - k = 0 such that α + β = 3 θ . \alpha + \beta = 3\theta.

Find k θ \dfrac{k}{\theta} .


The answer is 12.

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Guilherme Dela Corte by Guilherme Dela Corte , 24, Brazil

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1 solution

From Vieta's Formulas , we know that

α + β + θ = 12 \alpha + \beta + \theta = 12

But α + β = 3 θ \alpha + \beta = 3 \theta , so ( 3 θ ) + θ = 12 ↔ θ = 3 (3 \theta ) + \theta = 12 \leftrightarrow \theta = 3 .

Plugging x = 3 x = 3 back into the cubic equation generates us 3 3 − 12 ⋅ 3 2 + 39 ⋅ 3 − k = 0 ↔ 36 − k = 0 ↔ k = 36 3^3 - 12 \cdot 3 ^2 + 39 \cdot 3 - k = 0 \leftrightarrow 36 - k = 0 \leftrightarrow k = 36 .

Thus, k θ = 36 3 = 12. \dfrac{k}{\theta} = \dfrac{36}{3} = \boxed{12.}

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