(x+2/x-3)(x-2/x-3) Simplify

2 min read Jun 16, 2024

Simplifying the Expression (x + 2/x - 3)(x - 2/x - 3)

This expression involves fractions within parentheses, so we'll need to use some algebraic techniques to simplify it. Here's how we can break down the process:

1. Combining terms within each parenthesis:

We can rewrite the expression as:

(x² + 2/x - 3x - 6/x ) (x² - 2/x - 3x + 6/x )

This is done by applying the distributive property for each term within the parentheses.

2. Simplifying further:

Now we can combine the terms within each parenthesis:

(x² - 3x - 4/x) (x² - 3x + 4/x)

3. Recognizing the pattern:

We notice a pattern in the simplified expression. The first parenthesis is the same as the second, except the signs of the terms involving 'x' are different. This suggests we can use the 'difference of squares' pattern:

(a + b)(a - b) = a² - b²

4. Applying the difference of squares:

Let's consider:

a = x² - 3x
b = 4/x

Applying the pattern:

(x² - 3x + 4/x)(x² - 3x - 4/x) = (x² - 3x)² - (4/x)²

5. Final Simplification:

Expanding the squares:

(x⁴ - 6x³ + 9x² - 16/x²)

Therefore, the simplified expression for (x + 2/x - 3)(x - 2/x - 3) is x⁴ - 6x³ + 9x² - 16/x².