Final answer:
The value of the constant K is -5.2.
Step-by-step explanation:
We can use the factor theorem to find the value of the constant K. According to the factor theorem, if a polynomial f(x) has a remainder of c when divided by the binomial (ax - b), then f(b/a) = c. In this case, the polynomial is 4x³ + kx² + 7x - 23 and the binomial is 2x - 5. So, we set f(5/2) = 7 and solve for K:
f(5/2) = 4(5/2)³ + k(5/2)² + 7(5/2) - 23 = 7
Simplifying this equation:
(125/2) + (25/2)*k + (35/2) - 23 = 7
125/2 + (25/2)*k + 35/2 - 46/2 = 7
(125 + 25k + 35 - 46)/2 = 7
(144 + 25k)/2 = 7
144 + 25k = 14
25k = -130
k = -5.2