Hey Mila C.,
Here's my explanation to your question.
Given that (x-2) has a remainder of 10, we can also say that when x = 2, P(x) = 10. We know this as it is stated in the Polynomial remainder theorem. Using this information we can plug in 2 where there exists x's in our function and we can set that equal to 10.
- P(x) = x^4-3x^2+kx-2
- P(x) = 10 = (2)^4-3(2)^2+k(2)-2
Now we have worked our way down to one variable: k. Let's simplify further.
- 10 = 4+k(2)-2
- 10 = 2+k(2)
- 8 = k(2)
- 4 = k
If you have any further questions, I would be happy to help.