(4x ^ 5 - 25x ^ 4 + 40x ^ 3 + 5x ^ 2 - 30x - 18)/(x ^ 2 - 6x + 9)

We are tasked with simplifying the rational expression:

\[
\frac{4x^5 - 25x^4 + 40x^3 + 5x^2 - 30x - 18}{x^2 - 6x + 9}
\]

### Step 1: Factor the denominator

First, notice that the denominator \(x^2 - 6x + 9\) is a perfect square trinomial:

\[
x^2 - 6x + 9 = (x - 3)^2
\]

### Step 2: Analyze the numerator

Now, we need to factor or simplify the numerator. We will try polynomial division of:

\[
4x^5 - 25x^4 + 40x^3 + 5x^2 - 30x - 18
\]

by

\[
x^2 - 6x + 9
\]

I'll proceed with the polynomial division to check for simplifications.After performing polynomial division, the result is:

\[
\frac{4x^5 - 25x^4 + 40x^3 + 5x^2 - 30x - 18}{x^2 - 6x + 9} = 4x^3 - x^2 - 2x + 2 + \frac{-36}{x^2 - 6x + 9}
\]

So, the simplified form is:

\[
4x^3 - x^2 - 2x + 2 + \frac{-36}{(x - 3)^2}
\]

This represents the quotient \(4x^3 - x^2 - 2x + 2\) and a remainder term \(\frac{-36}{(x - 3)^2}\).

Would you like further details or clarification?

Here are 5 related questions:
1. How do we verify if a polynomial is divisible by another?
2. What are some methods for factoring higher-degree polynomials?
3. How does polynomial long division differ from synthetic division?
4. What is the significance of remainder in polynomial division?
5. How can we interpret the remainder in terms of rational functions?

**Tip:** Always check for special factorization patterns, like perfect squares or cubes, to simplify polynomial expressions.

The query submitted by the user: (4x^5 - 25x^4 + 40x^3 + 5x^2 - 30x - 18)/(x^2 - 6x + 9)

Discover how to simplify the rational expression (4x^5 - 25x^4 + 40x^3 + 5x^2 - 30x - 18)/(x^2 - 6x + 9) using polynomial division and factoring. Learn key algebraic concepts like polynomial long division and perfect square trinomials, suitable for high school students in Grades 10-12.

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