For what value of a is the polynomial (2x3 + ax2 + 11x + a + 3) exactly divisible by (x + 3)

To find the value of 'a' for which the polynomial (2x^3 + ax^2 + 11x + a + 3) is exactly divisible by (x + 3), we can use the Polynomial Remainder Theorem. According to this theorem, a polynomial f(x) is divisible by (x - c) if and only if f(c) = 0.

In this case, we want the polynomial to be divisible by (x + 3), which is the same as (x - (-3)). So, we set x = -3 in the polynomial and solve for 'a':

2*(-3)^3 + a*(-3)^2 + 11*(-3) + a + 3 = 0 -54 + 9a - 33 + a + 3 = 0 10a - 84 = 0 10a = 84 a = 84 / 10 a = 8.4

So, the polynomial is exactly divisible by (x + 3) when a = 8.4.