Decomposing Rational Expressions into Partial Fractions: A Step-by-Step Guide

In this article, we will guide you through the process of decomposing the rational expression ( frac{x^5 - 2x^4 x^3 - x 5}{x^3 - 2x^2 - x - 2}) into partial fractions. This involves several steps, including polynomial long division, factoring the denominator, and solving for the coefficients of the partial fractions.

Step 1: Polynomial Long Division

The first step is to perform polynomial long division because the degree of the numerator is greater than the degree of the denominator. Here are the steps:

Divide the leading term of the numerator by the leading term of the denominator: Multiply the entire denominator by the result from Step 1. Subtract this result from the original numerator.

Step 1:

Divide the leading term of the numerator by the leading term of the denominator:

( frac{x^5}{x^3} x^2)

Step 2:

Multiply the entire denominator by x^2:

x^2(x^3 - 2x^2 - x - 2) x^5 - 2x^4 - x^3 - 2x^2

Step 3:

Subtract this result from the original numerator:

(x^5 - 2x^4 - x^3 - x 5) - (x^5 - 2x^4 - x^3 - 2x^2) 2x^2 - x 5

Our expression now becomes:

( frac{x^5 - 2x^4 - x^3 - x 5}{x^3 - 2x^2 - x - 2} x^2 frac{2x^2 - x 5}{x^3 - 2x^2 - x - 2})

Step 2: Factoring the Denominator

The next step is to factor the denominator. We can factor it into simpler terms using the Rational Root Theorem, which helps us find potential roots.

Testing (x 2) (since (2^3 - 2*2^2 - 2 - 2 0))

Thus, (x - 2) is a factor, and we can perform synthetic division with the remaining polynomial.

Synthetic Division

When performing synthetic division:

2 1 -2 1 -2 2 0 2 1 0 1 0

So the factored form of the denominator is:

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

Step 3: Setting Up Partial Fractions

The next step is to express the remaining fraction in terms of partial fractions. We set up the partial fraction decomposition as follows:

( frac{2x^2 - x 5}{(x - 2)(x - 1)(x 1)} frac{A}{x - 2} frac{Bx C}{x^2 - 1} )

Step 4: Solving for A, B, and C

Multiplying both sides by the original denominator:

(x - 2)(x - 1)(x 1)(2x^2 - x 5) A(x - 1)(x 1) (Bx C)(x - 2))

Expanding and equating coefficients:

1) A B 2 2) -2B C -1 3) A - 2C 5

Solving this system of equations:

From A B 2, B 2 - A Substituting B 2 - A into -2B C -1, we get -2(2 - A) C -1, -4 2A C -1, hence C 3 - 2A Substituting C 3 - 2A into A - 2C 5, we get A - 2(3 - 2A) 5, A - 6 4A 5, 5A 11, so A 3 Substituting A 3 into B 2 - A, we get B -1 Substituting A 3 into C 3 - 2A, we get C -3

Thus the partial fraction decomposition is:

( frac{2x^2 - x 5}{(x - 2)(x - 1)(x 1)} frac{3}{x - 2} frac{-x - 1}{x^2 - 1})

Combining All Steps

Putting it all together, the complete decomposition of the original expression is:

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