Question

What value does k hold if the polynomial x^3 + kx^2 + 7x + 5, when divided by x + 6, leaves a remainder of -1?

A. 5
B. -7
C. 7
D. -5

Answer 1

To find the value of k in the polynomial x^3 + kx^2 + 7x + 5, we use the Remainder Theorem. By substituting x with -6 and setting the result equal to -1 (the remainder), we calculate k to be 7.

Step-by-step explanation:

To determine the value of k in the polynomial x^3 + kx^2 + 7x + 5, given that it leaves a remainder of -1 when divided by x + 6, we can use the Remainder Theorem. This theorem states that if a polynomial f(x) is divided by x - c, the remainder is f(c). Applying this to our case, we set c = -6 because we are dividing by x + 6.

Then we calculate:

f(-6) = (-6)^3 + k(-6)^2 + 7(-6) + 5
= -216 + 36k - 42 + 5
= 36k - 253

Since we know the remainder is -1, we set the expression equal to -1 and solve for k:

36k - 253 = -1
36k = 252
k = 7

Therefore, the value of k is 7, which corresponds to option C.