To solve the given expression \(\frac{8y^3 - 36y^2 + 54 - 27}{2y - 3}\), we need to perform polynomial division.

Let's rewrite the numerator and the denominator:
Numerator: \(8y^3 - 36y^2 + 54 - 27\)
Denominator: \(2y - 3\)

First, we'll set up the division using polynomial long division:

1. Divide the first term of the numerator by the first term of the denominator: \(\frac{8y^3}{2y} = 4y^2\).
2. Multiply the entire denominator \(2y - 3\) by \(4y^2\): \((2y - 3) \cdot 4y^2 = 8y^3 - 12y^2\).
3. Subtract this result from the original numerator:
   \[
   (8y^3 - 36y^2 + 54 - 27) - (8y^3 - 12y^2) = -24y^2 + 54 - 27
   \]

Next, we repeat the process with the new polynomial \(-24y^2 + 54 - 27\):

4. Divide the first term of the new polynomial by the first term of the denominator: \(\frac{-24y^2}{2y} = -12y\).
5. Multiply the entire denominator \(2y - 3\) by \(-12y\): \((2y - 3) \cdot -12y = -24y^2 + 36y\).
6. Subtract this result from the current polynomial:
   \[
   (-24y^2 + 54 - 27) - (-24y^2 + 36y) = 54y - 27
   \]

Continue with the polynomial \(54y - 27\):

7. Divide the first term of the new polynomial by the first term of the denominator: \(\frac{54y}{2y} = 27\).
8. Multiply the entire denominator \(2y - 3\) by \(27\): \((2y - 3) \cdot 27 = 54y - 81\).
9. Subtract this result from the current polynomial:
   \[
   (54y - 27) - (54y - 81) = 54
   \]

Thus, the final remainder is 54. Combining all the terms we found:

\[
\frac{8y^3 - 36y^2 + 54 - 27}{2y - 3} = 4y^2 - 12y + 27 + \frac{54}{2y - 3}
\]

So, the quotient is \(4y^2 - 12y + 27\) with a remainder of \(54\), and the expression is:

\[
4y^2 - 12y + 27 + \frac{54}{2y - 3}
\]

Would you like more details or have any questions about this solution?

Here are some related questions you might find interesting:

1. How do you perform polynomial long division?
2. What is synthetic division, and how does it differ from long division?
3. Can you explain the Remainder Theorem?
4. How can you verify the result of polynomial division?
5. What are some applications of polynomial division in calculus?
6. How do you find the roots of a polynomial?
7. What is the Factor Theorem?
8. Can polynomial division be used to simplify rational expressions?

**Tip:** Always check your work by multiplying the quotient by the divisor and adding the remainder to ensure you get back the original polynomial.

Learn how to solve the polynomial division problem involving 8y³ - 36y² + 54 - 27 divided by (2y - 3). Step-by-step solution provided, covering polynomial long division. Suitable for advanced high school to college students.

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