Expressing ( frac{x^2}{x^2-3x-2} ) in Partial Fractions

Understanding how to express algebraic fractions, specifically ( frac{x^2}{x^2-3x-2} ), in partial fractions is a crucial skill in algebra. This process helps break down complex expressions into simpler forms for easier manipulation and integration. Below, we'll explore the step-by-step process of expressing ( frac{x^2}{x^2-3x-2} ) in partial fractions.

Step-by-Step Guide

To express ( frac{x^2}{x^2-3x-2} ) in partial fractions, follow these steps:

1. Perform Polynomial Division

First, note that the numerator and the denominator have the same degree. Since this is the case, we must perform polynomial division. The quotient is 1 with a remainder of -3x-2. Therefore, we can write:

( frac{x^2}{x^2-3x-2} 1 frac{-3x-2}{x^2-3x-2} )

2. Simplify the Remainder

The remainder -3x-2 can be further split into -3x-2 3x-2-3x-4 for easier manipulation. Thus, we can write:

( frac{x^2}{x^2-3x-2} 1 - frac{3x-2}{x^2-3x-2} )

Now, we need to express (frac{3x-2}{x^2-3x-2}) in partial fractions. Notice that x^2-3x-2 can be factored into (x-1)(x-2). So, we write:

(frac{3x-2}{x^2-3x-2} frac{3x-2}{(x-1)(x-2)} frac{A}{x-1} frac{B}{x-2})

3. Find the Constants A and B

By comparing the coefficients, we find:

A(x-2) B(x-1) 3x - 2

Setting up the system of equations:

A B 3 (Coefficient of x)

-2A - B -2 (Constant term)

Solving these simultaneously, we get:

A -1, B 4

Thus, we can write:

(frac{3x-2}{(x-1)(x-2)} -frac{1}{x-1} frac{4}{x-2})

4. Combine the Fractions

Putting everything together, we get:

(frac{x^2}{x^2-3x-2} 1 - frac{1}{x-1} frac{4}{x-2})

Explanation and Verification

Finally, to verify our result, we can combine the fractions back:

1 - frac{1}{x-1} frac{4}{x-2} 1 - frac{4}{x-2} - frac{1}{x-1})

This can be simplified to:

1 - frac{4}{x-2} - frac{1}{x-1} frac{(x-1)(x-2) - 4(x-1) - (x-2)}{(x-1)(x-2)}

Simplifying the numerator:

(x-1)(x-2) - 4(x-1) - (x-2) x^2 - 3x 2 - 4x 4 - x 2 x^2 - 3x - 2

Thus, we confirm that:

(frac{x^2}{x^2-3x-2} 1 - frac{1}{x-1} frac{4}{x-2})

Additional Resources

For a more detailed explanation and additional examples, I suggest watching the following Khan Academy video:

Intro to Partial Fraction Expansion - Khan Academy

Understanding partial fractions is essential for solving more complex algebraic equations and is a fundamental skill in calculus. Whether you are a student or someone looking to refresh your algebra skills, this method is both practical and effective.