The Remainder Theorem states that if a polynomial f(x) is divided by (x - c), then the remainder is f(c). This means that if the polynomial f(x) = 2x^2 + kx + 9 is divided by (x - 4) and the remainder is 81, then we can find the value of k by setting f(4) equal to 81 and solving for k.
So:
f(4) = 81
2(4)^2 + k(4) + 9 = 81
2(16) + 4k + 9 = 81
32 + 4k + 9 = 81
4k + 41 = 81
4k = 81 - 41
4k = 40
k = 40 / 4
k = 10
So, the value of k is 10.