We are given the polynomial $P(x) = 4x^3 - 4x + 1$. We want to find the remainder when $P(x)$ is divided by $2x - 3$.

1. Problem Description

We are given the polynomial $P(x) = 4x^3 - 4x + 1$. We want to find the remainder when $P(x)$ is divided by $2x - 3$.

2. Solution Steps

By the Remainder Theorem, if we divide a polynomial $P(x)$ by $ax - b$, the remainder is $P(\frac{b}{a})$.

In this case, we are dividing by $2x - 3$, so $a = 2$ and $b = 3$. Therefore, we need to find $P(\frac{3}{2})$.

We have:

$P(x) = 4x^3 - 4x + 1$

$P(\frac{3}{2}) = 4(\frac{3}{2})^3 - 4(\frac{3}{2}) + 1$

$P(\frac{3}{2}) = 4(\frac{27}{8}) - 4(\frac{3}{2}) + 1$

$P(\frac{3}{2}) = \frac{27}{2} - 6 + 1$

$P(\frac{3}{2}) = \frac{27}{2} - 5$

$P(\frac{3}{2}) = \frac{27}{2} - \frac{10}{2}$

$P(\frac{3}{2}) = \frac{17}{2}$

3. Final Answer

$P(\frac{3}{2}) = \frac{17}{2}$