Decomposing a Rational Expression into Partial Fractions: A Step-by-Step Guide

In algebra, decomposing a rational expression into partial fractions is a crucial skill that helps simplify complex fractions. This process involves expressing a rational expression as the sum of simpler fractions. Today, we will guide you through the steps to decompose the expression (frac{5x - 1}{x^2(x^2 - 4)}) into partial fractions.

Step 1: Write the Equation

The first step is to write the expression in the form of partial fractions. Given the rational expression (frac{5x - 1}{x^2(x^2 - 4)}), our goal is to express it in the form:

(frac{5x - 1}{x^2(x^2 - 4)} frac{A}{x} frac{B}{x^2} frac{Cx D}{x^2 - 4})

Step 2: Combine the Right Side

Now, we need to combine the right-hand side to have a common denominator. Therefore, we rewrite the equation as:

(frac{5x - 1}{x^2(x^2 - 4)} frac{Axx^2 - 4 Bx^2 - 4 Cx Dx^2}{x^2(x^2 - 4)})

Step 3: Set the Numerators Equal

Since the denominators are the same, we can set the numerators equal to each other:

(5x - 1 A(x^2 - 4) Bx^2 - 4 Cx Dx^2)

Step 4: Expand the Right Side

Expanding the right-hand side, we get:

(5x - 1 Ax^2 - 4A Bx^2 - 4 Cx Dx^2)

Simplifying further, we have:

(5x - 1 (A B D)x^2 Cx - 4A - 4)

Step 5: Equate the Coefficients

To find the values of A, B, C, and D, we need to equate the coefficients of like terms:

(A B D 0) (C 5) (-4A - 4 -1)

Step 6: Solve the Equations

Let's solve for A, B, C, and D:

From (-4A - 4 -1), we find: (-4A 3) (A frac{3}{-4}) From (C 5), we find: (C 5) Substituting (A frac{3}{-4}) into the equation (A B D 0), we find: (-frac{3}{4} B D 0) From (B frac{1}{4}) and substituting into the equation, we find: (-frac{3}{4} frac{1}{4} D 0) (D frac{2}{4}) or (D frac{1}{2})

Step 7: Write the Final Result

Therefore, the values of A, B, C, and D are:

(A frac{3}{-4}), (B frac{1}{4}), (C 5), (D frac{1}{2})

Substituting these values back into the partial fraction decomposition, we get:

(frac{5x - 1}{x^2(x^2 - 4)} frac{3}{-4x} frac{1}{4x^2} frac{5x frac{1}{2}}{x^2 - 4})

Conclusion

By carefully applying these steps, we can decompose complex rational expressions into simpler partial fractions. This technique is invaluable in solving algebraic equations, simplifying integrals, and analyzing functions in calculus.

Additional Notes

Understanding partial fractions is essential for advanced topics in algebra and calculus. If you come across similar problems, you can follow these systematic steps to solve them accurately.