The number of positive integers $x$ that satisfy $3^x = x^3 + 3x^2 + 2x + 1$ is:
Step-by-Step Solution
Key Concept: Recognize that $x^3 + 3x^2 + 2x + 1 = x(x+1)(x+2) + 1$ by polynomial factorization. Then analyze divisibility properties: $3^x \equiv 0 \pmod{3}$ while $x(x+1)(x+2) + 1 \equiv 1 \pmod{3}$ for all integers $x$, creating a modular arithmetic contradiction.
We have $3^x = x(x+1)(x+2) + 1$. The LHS is a multiple of 3 while the RHS is not divisible by 3, creating a contradiction.
Correct Answer: 1