Question #40353

This question tests the Quadrant Adjustment Logic and extraneous solution rejection in trigonometric polynomial equations.

Students typically map the polynomial to a multiple-angle identity (Chebyshev polynomial) and solve the algebraic equation $16x^5 - 20x^3 + 5x = 2x^2 - 1$, yielding five roots.

The critical trap lies in failing to apply the principal value range of the inverse cosine function.

Because the LHS $\cos^{-1}(...)$ is strictly bounded in $[0, \pi]$, the RHS $2\cos^{-1}x$ must also lie in $[0, \pi]$, which enforces the hidden domain restriction $x \ge 0$.

Students who ignore this will incorrectly sum all algebraic roots (yielding 0) or fail to reject the negative roots $\cos(4\pi/7)$ and $\cos(6\pi/7)$.