Step-by-step Solution:

The equations correspond to \( P(t) = xt^2 + yt^4 + zt^6 \), but notice a pattern in the coefficients. Let's form an algebraic identity or solve it as a linear system:\n\[ x + y + z = 1 \]\n\[ 4x + 9y + 16z = 25 \]\n\[ 16x + 81y + 256z = 625 \]\n\nBy Cramers rule or substitution:\nFrom (1), \( z = 1 - x - y \).\nSubstitute \( z \) into (2):\n\[ 4x + 9y + 16(1 - x - y) = 25 \implies -12x - 7y = 9 \implies 12x + 7y = -9 \quad \text{--- (Eq 4)} \]\n\nSubstitute \( z \) into (3):\n\[ 16x + 81y + 256(1 - x - y) = 625 \implies -240x - 175y = 369 \quad \text{--- (Eq 5)} \]\n\nFrom (Eq 4), \( y = \frac{-9 - 12x}{7} \).\nSubstitute into (Eq 5):\n\[ -240x - 175\left(\frac{-9 - 12x}{7}\right) = 369 \]\n\[ -240x - 25(-9 - 12x) = 369 \]\n\[ -240x + 225 + 300x = 369 \]\n\[ 60x = 144 \]\n\[ x = \frac{144}{60} = \frac{36}{15} = \frac{12}{5} \]\n\nThus, \( x = \frac{36}{15} \).