Final answer:
To find the value of k for which x-4 is a factor of the polynomial x³-kx²+2kx-12, we use the Remainder Theorem and substitute x with 4. Solving the resulting equation, we find that k must be 6.5.
Step-by-step explanation:
The student asked to find the value of k such that x-4 is a factor of x³-kx²+2kx-12. To find k, we can use the Remainder Theorem, which states that if x-a is a factor of a polynomial, then the polynomial will equal zero when x is replaced by a. Hence, we replace x with 4 in the polynomial:
4³-k·4²+2k×4-12 = 0.
Simplifying this, we get 64-16k+8k-12 = 0, then simplifying further gives 52=8k. Dividing both sides by 8, we arrive at k = 6.5.