Final answer:
To determine the value of d when (x - 5) is a factor of y = x^4 + dx^3 + 2x^2 - 50, we apply the factor theorem, setting x equal to 5 and solving for d, which in this case is -5.
Step-by-step explanation:
If (x - 5) is a factor of the polynomial y = x^4 + dx^3 + 2x^2 - 50, we can find the value of d by using the factor theorem. The factor theorem states that if (x - c) is a factor of a polynomial, then the polynomial will equal zero when x = c. In this case, we substitute x = 5 into the polynomial and set it equal to zero.
Substitute x = 5:
5^4 + d(5^3) + 2(5^2) - 50 = 0
625 + 125d + 50 - 50 = 0
625 + 125d = 0
Solve for d:
125d = -625
d = -5
Therefore, the value of d is -5.