Answer:
Explanation:
To determine the value of k such that (x-4) is a factor of the polynomial f(x) = x³ + 2x² - 11x + k, we need to find the remainder when f(x) is divided by (x-4). If the remainder is zero, then (x-4) is a factor of the polynomial.
Using polynomial long division, we divide f(x) by (x-4):
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x² + 6x + 5
____________________
x - 4 | x³ + 2x² - 11x + k
- (x³ - 4x²)
___________
6x² - 11x
- (6x² - 24x)
___________
13x + k
- (13x - 52)
___________
k + 52
The remainder is k + 52. For (x-4) to be a factor of the polynomial, the remainder should be zero. Therefore, we have the equation k + 52 = 0.
Solving for k, we get:
k = -52
So, the value of k that makes (x-4) a factor of the polynomial is k = -52.