1. **State the problem:** We are given the profit function of a small business: $$P(t) = 0.5t^4 - 3t^3 + t^2 + 25$$ where $t$ is the number of years since 1990. We need to find the profit in 2017. 2. **Identify the value of $t$ for 2017:** Since $t$ counts years since 1990, $$t = 2017 - 1990 = 27$$ 3. **Use the Remainder Theorem:** The Remainder Theorem states that the remainder when a polynomial $P(t)$ is divided by $(t - a)$ is $P(a)$. Here, to find the profit at $t=27$, we evaluate $P(27)$. 4. **Calculate $P(27)$:** $$P(27) = 0.5(27)^4 - 3(27)^3 + (27)^2 + 25$$ Calculate powers: $$27^2 = 729$$ $$27^3 = 27 \times 729 = 19683$$ $$27^4 = 27 \times 19683 = 531441$$ Substitute: $$P(27) = 0.5 \times 531441 - 3 \times 19683 + 729 + 25$$ Calculate each term: $$0.5 \times 531441 = 265720.5$$ $$3 \times 19683 = 59049$$ So: $$P(27) = 265720.5 - 59049 + 729 + 25$$ Simplify stepwise: $$265720.5 - 59049 = 206671.5$$ $$206671.5 + 729 = 207400.5$$ $$207400.5 + 25 = 207425.5$$ 5. **Interpret the result:** The profit in 2017 is $207425.5$ thousand dollars, or 207,425,500 dollars. **Final answer:** $$\boxed{207425.5}$$ (in thousands of dollars)