When *P*(*x*) = 4*x*3 – 4*x* + 1 is divided by 2*x* – 3, the remainder is

To find the remainder when \( P(x) = 4x^3 - 4x + 1 \) is divided by \( 2x - 3 \), we can use the Remainder Theorem. According to this theorem, the remainder of the polynomial \( P(x) \) when divided by \( ax - b \) is given by \( P\left(\frac{b}{a}\right) \).

In this case:
- \( a = 2 \)
- \( b = 3 \)

First, we calculate \( \frac{b}{a} \):

\[
\frac{3}{2} = 1.5
\]

Now we evaluate \( P\left(\frac{3}{2}\right) \):

\[
P\left(\frac{3}{2}\right) = 4\left(\frac{3}{2}\right)^3 - 4\left(\frac{3}{2}\right) + 1
\]

Calculating each term:

1. \( \left(\frac{3}{2}\right)^3 = \frac{27}{8} \)
2. \( 4 \left(\frac{27}{8}\right) = \frac{108}{8} = 13.5 \)
3. \( 4 \left(\frac{3}{2}\right) = 6 \)

Now, substitute back into \( P\left(\frac{3}{2}\right) \):

\[
P\left(\frac{3}{2}\right) = 13.5 - 6 + 1 = 8.5
\]

Thus, the remainder when \( P(x) \) is divided by \( 2x - 3 \) is:

\[
\boxed{8.5}
\]

### Do you want details or have any questions? Here are some related questions:

1. What would happen if we used synthetic division instead?
2. Can you explain the Remainder Theorem in more detail?
3. How can we find the quotient when dividing \( P(x) \) by \( 2x - 3 \)?
4. What is the significance of the remainder in polynomial division?
5. How would the remainder change if the divisor were \( x - 1 \) instead? 

**Tip:** Always double-check your polynomial evaluations by substituting back to confirm calculations.

When P(x) = 4*x^3 – 4*x + 1 is divided by 2*x – 3, the remainder is

Discover how to find the remainder when dividing the polynomial P(x) = 4x^3 – 4x + 1 by 2x – 3. This detailed solution uses the Remainder Theorem and provides step-by-step calculations to arrive at the result of 8.5. Ideal for high school students studying algebra.

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