Final answer:
The value of c that makes the polynomial p(x) = cx³ - 15x - 68 divisible by (x-4) is determined using the Remainder Theorem. By plugging in x = 4 and setting the equation to zero, we solve for c, which yields c = 2.
Step-by-step explanation:
To find the value of c so that the polynomial p(x) is divisible by (x-4), we can use the Remainder Theorem. The Remainder Theorem states that if a polynomial f(x) is divided by (x-a), the remainder is f(a). In this case, for p(x) to be divisible by (x-4), p(4) must equal zero.
So, we substitute x = 4 into p(x) = cx³ - 15x - 68 and set the expression equal to zero:
p(4) = c(4)^3 - 15(4) - 68 = 0, which simplifies to
64c - 60 - 68 = 0,
Now, we combine like terms and solve for c:
64c - 128 = 0
64c = 128
c = 2
Therefore, the value of c that makes p(x) divisible by (x-4) is 2.