Final answer:
The remainder when the polynomial f(x) = 2^3 - 5^2 - 5^2 + 5 is divided by x + 2 is evaluated by substituting x with -2, resulting in a remainder of -37.
Step-by-step explanation:
The remainder when f(x) = 23 - 52 - 52 + 5 is divided by x + 2 is determined by evaluating f(-2), since the remainder theorem states that the remainder of a polynomial f(x) divided by x - a is f(a). First, we calculate the exponents and operations within the function: 23 equals 8, and 52 equals 25. Therefore, the function simplifies to 8 - 25 - 25 + 5. When we add and subtract the terms, we get -37. Thus, when we evaluate f(-2), we substitute x with -2 in the polynomial: 8 - 25 - 25 + 5 = -37.
However, if we are dividing by x + 2, we would evaluate f(-2):
- f(-2) = 8 - (25) - (25) + 5
- f(-2) = 8 - 50 + 5
- f(-2) = -42 + 5
- f(-2) = -37
Hence, the remainder when f(x) is divided by x + 2 is -37.