To find the remainder when \( f(x) = 3x^5 - 4x + 1 \) is divided by \( x + 1 \), we can use the Remainder Theorem. According to the Remainder Theorem, the remainder of the division of a polynomial \( f(x) \) by \( x - c \) is \( f(c) \).
In this case, we are dividing by \( x + 1 \), which can be rewritten as \( x - (-1) \). Therefore, we need to evaluate \( f(-1) \):
\[ f(-1) = 3(-1)^5 - 4(-1) + 1 \] \[ = 3(-1) - 4(-1) + 1 \] \[ = -3 + 4 + 1 \] \[ = 2 \]
Thus, the remainder when \( f(x) \) is divided by \( x + 1 \) is \( 2 \). The answer is:
2