To find the value of k, if x-1 is a factor of the polynomial 4x^3 + 3x^2 - 4x - k, we need to use the factor theorem. The factor theorem states that if a polynomial f(x) has a factor (x-a), then f(a) = 0.

Let's apply the factor theorem to this problem.

Step 1: Determine the polynomial f(x)
The given polynomial is 4x^3 + 3x^2 - 4x - k

Step 2: Substitute x = 1 into the polynomial
To check if x-1 is a factor, we substitute x = 1 into the polynomial and see if the result is equal to 0.

f(1) = 4(1)^3 + 3(1)^2 - 4(1) - k
= 4 + 3 - 4 - k
= 3 - k

Step 3: Set f(1) = 0 and solve for k
Since x-1 is a factor, f(1) should be equal to 0.

3 - k = 0
k = 3

Therefore, the value of k is 3.

Explanation:
We used the factor theorem to determine the value of k. The factor theorem states that if a polynomial has a factor (x-a), then f(a) = 0. By substituting x = 1 into the polynomial, we obtained f(1) = 3 - k. Since we want f(1) to be equal to 0, we set 3 - k = 0 and solved for k, which gives us k = 3. Hence, the value of k is 3.

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