Minimal Polynomial Stuff 2

Geometry Level 4

Find the (monic) minimal polynomial of tan ⁡ π 7 \tan \frac{\pi}7 with rational coefficients. Submit the coefficient of x 2 x^2 .


The answer is 35.

Christopher Criscitiello by Christopher Criscitiello , 25, USA

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1 solution

From tan ⁡ ( π − θ ) = − tan ⁡ θ \tan (\pi - \theta) = - \tan \theta , we have

tan ⁡ ( π − 3 π 7 ) = − tan ⁡ 3 π 7 Let x = tan ⁡ π 7 4 x 1 − x 2 1 − 4 x 2 ( 1 − x 2 ) 2 = − x + 2 x 1 − x 2 1 − 2 x 2 1 − x 2 4 x ( 1 − x 2 ) x 4 − 6 x 2 + 1 = − x ( 3 − x 2 ) 1 − 3 x 2 4 ( 1 − x 2 ) ( 3 x 2 − 1 ) = ( 3 − x 2 ) ( x 4 − 6 x 2 + 1 ) x 6 − 21 x 4 + 35 x 2 − 7 = 0 \begin{aligned} \tan \left(\pi - \frac {3\pi}7\right) & = - \tan \frac {3\pi}7 & \small \color{#3D99F6} \text{Let }x = \tan \frac \pi 7 \\ \frac {\frac {4x}{1-x^2}}{1-\frac {4x^2}{(1-x^2)^2}} & = - \frac {x+\frac {2x}{1-x^2}}{1-\frac {2x^2}{1-x^2}} \\ \frac {4x(1-x^2)}{x^4-6x^2+1} & = - \frac {x(3-x^2)}{1-3x^2} \\ 4(1-x^2)(3x^2-1) & = (3-x^2)(x^4-6x^2+1) \\ x^6 - 21x^4+\boxed{35}x^2-7 & = 0 \end{aligned}

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