5. If \( k \) is a constant, what is the value of \( k \) such that the polynomial \( k^{2} x^{3}+4 k x+4 \) is divisible by \( (x-1) \) ?

To determine the value of \( k \) that makes the polynomial \( k^{2} x^{3}+4 k x+4 \) divisible by \( (x-1) \), we can use the Remainder Theorem. According to this theorem, a polynomial \( P(x) \) is divisible by \( (x-c) \) if \( P(c) = 0 \). In this case, we need to evaluate \( P(1) \): \[ P(1) = k^{2}(1)^{3} + 4k(1) + 4 = k^{2} + 4k + 4 \] Setting \( P(1) = 0 \): \[ k^{2} + 4k + 4 = 0 \] This can be factored as: \[ (k + 2)^{2} = 0 \] Thus, \( k + 2 = 0 \), giving us \( k = -2 \). So, the value of \( k \) such that the polynomial is divisible by \( (x-1) \) is \( k = -2 \).