Final answer:
To find the value of k in the given polynomial divided by x-4 and resulting in a remainder of -7, we can use polynomial long division. The value of k is -17.
Step-by-step explanation:
To find the value of k, we can use polynomial long division to divide the given polynomial by x-4. This will give us the quotient and the remainder. Since the remainder is given as -7, we can set it equal to -7 and solve for k.
Dividing x^3 + kx^2 + 7x + 5 by x-4:
(x^2 + (4 + k)x + (16 + 4k)) + (61 + 4k)/(x-4)
Since the remainder is -7, we have:
(61 + 4k) = -7
Now, solve for k:
4k = -7 - 61
4k = -68
k = -68/4
k = -17