(X 2)/(X-2)+(X 3)/(X-3)=(X-2)/(X 2)+(X-3)/(X 3)

Let's solve the given equation step-by-step. The equation is:

\[
\frac{X^2}{X - 2} + \frac{X^3}{X - 3} = \frac{X - 2}{X^2} + \frac{X - 3}{X^3}
\]

### Step 1: Simplify the Left-Hand Side (LHS)

First, we simplify each term on the left-hand side of the equation.

\[
\frac{X^2}{X - 2} + \frac{X^3}{X - 3}
\]

No common denominator is immediately apparent, so we'll keep it as is for now.

### Step 2: Simplify the Right-Hand Side (RHS)

Next, we simplify each term on the right-hand side of the equation.

\[
\frac{X - 2}{X^2} + \frac{X - 3}{X^3}
\]

No common denominator here either, so we'll also keep it as is.

### Step 3: Cross Multiplication

To eliminate the fractions, let's cross-multiply. Multiply the numerator of each side by the denominator of the other:

\[
(X^2)(X^3) + (X^3)(X - 2) = (X^2)(X - 3) + (X^3)(X - 2)
\]

Simplify this equation:

\[
X^5 + X^3(X - 2) = X^2(X - 3) + X^3(X - 2)
\]

### Step 4: Expand Each Term

Now expand each term:

\[
X^5 + X^4 - 2X^3 = X^3 - 3X^2 + X^4 - 2X^3
\]

### Step 5: Combine Like Terms

Combine like terms on both sides:

\[
X^5 + X^4 - 2X^3 = X^4 - 2X^3 + X^3 - 3X^2
\]

Subtract \(X^4 - 2X^3\) from both sides:

\[
X^5 = X^3 - 3X^2
\]

### Step 6: Factor Out Common Terms

Factor out \(X^2\) on the right-hand side:

\[
X^5 = X^2(X - 3)
\]

Now divide both sides by \(X^2\) (assuming \(X \neq 0\)):

\[
X^3 = X - 3
\]

### Step 7: Move All Terms to One Side

Move all terms to one side to set the equation to zero:

\[
X^3 - X + 3 = 0
\]

### Step 8: Find the Roots

Finding the roots of a cubic equation can be complex. However, let's analyze whether this cubic equation has simple roots.

After some inspection or by using numerical methods or software, we find that there is one real root and two complex roots for this cubic polynomial.

Given the nature of this equation, we need to solve \(X^3 - X + 3 = 0\) to get the exact roots.

### Conclusion

The roots of the equation \(\frac{X^2}{X - 2} + \frac{X^3}{X - 3} = \frac{X - 2}{X^2} + \frac{X - 3}{X^3}\) are determined by solving the cubic polynomial \(X^3 - X + 3 = 0\).

Would you like a detailed step-by-step explanation for finding these roots, or do you have any specific questions?

### Relative Questions

1. How do you find the roots of a cubic polynomial like \(X^3 - X + 3 = 0\)?
2. What methods are available for solving cubic equations?
3. Are there any real-world applications for equations like this?
4. What is the significance of cross-multiplication in solving rational equations?
5. How do you simplify expressions involving fractions with polynomials?

### Tip

When working with equations that involve polynomials in the numerator and denominator, always consider factoring as a first step to simplify the equation.

Learn how to solve the rational equation X^2/(X-2) + X^3/(X-3) = (X-2)/X^2 + (X-3)/X^3 with a detailed step-by-step solution. This problem involves rational equations, cross-multiplication, and cubic equations. Suitable for high school students.

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