Final answer:
To find the value of k such that (x^3−x^2−kx−36)÷(x+2) has a remainder of zero, we can use the remainder theorem. By setting (x+2)=0 and solving for x, we find x=-2. Substituting x=-2 into the polynomial (x^3−x^2−kx−36) and solving for k, we find k=24.
Step-by-step explanation:
To find the value of k such that (x^3−x^2−kx−36)÷(x+2) has a remainder of zero, we need to divide the polynomial (x^3−x^2−kx−36) by (x+2) and find the value of k for which the remainder is zero. To do this, we can use the remainder theorem. According to the remainder theorem, if the polynomial (x^3−x^2−kx−36) divided by (x+2) has a remainder of zero, then (x+2) is a factor of the polynomial.
Therefore, we can set (x+2)=0 and solve for x:
x+2=0
x=-2
Now, substitute x=-2 into the polynomial (x^3−x^2−kx−36) and solve for k:
(-2)^3−(-2)^2−k(-2)−36=0
-8-4+2k-36=0
2k-48=0
2k=48
k=24
Therefore, the value of k such that (x^3−x^2−kx−36)÷(x+2) has a remainder of zero is k=24.