Final answer:
To find the non-zero values of "a" for the polynomial's same remainder condition, we employ the Remainder Theorem, setting up an equation using the values f(a) and f(-12a) and solving for "a".
Step-by-step explanation:
We are tasked with finding the non-zero values of "a" for which the polynomial x^3 - 5x^2 - 2x + 24 gives the same remainder when divided by x - a and x + 12a. To do this, we will use the Remainder Theorem, which states that if a polynomial f(x) is divided by x - r, the remainder is f(r).
Firstly, for x - a, let's evaluate the polynomial at a:
f(a) = a^3 - 5a^2 - 2a + 24
For x + 12a, let's evaluate the polynomial at -12a:
f(-12a) = (-12a)^3 - 5(-12a)^2 - 2(-12a) + 24
Now, in order for these two remainders to be equal, we must have:
f(a) = f(-12a)
Which leads to an equation we must solve for a. The solution to this equation will provide the non-zero values of a that meet the condition.