How to Decompose 7x 18 / (x^2 x - 3) into Partial Fractions

Partial fraction decomposition is a powerful algebraic technique for breaking down complex rational expressions into simpler, more manageable parts. This article will guide you through the step-by-step process of decomposing the expression 7x 18 / (x^2 x - 3) into partial fractions. We will use the specific example to explain the general method of partial fraction decomposition for rational expressions with quadratic denominators.

Introduction to Partial Fraction Decomposition

Partial fractions are used to break down rational expressions into simpler components. This is particularly useful when dealing with integrals, simplifying complex fractions, and solving differential equations. For polynomials with linear factors, the numerators of the partial fractions are constants. However, when polynomials have quadratic factors, the numerators can be linear. In this case, our expression 7x 18 / (x^2 x - 3) requires a bit more attention.

Step-by-Step Decomposition

The first step is to factor the denominator if possible. Here, we identify the factors of x^2 x - 3 as (x 2)(x - 1). Therefore, we can write:

``` (7x 18) / (x^2 x - 3) A / (x 2) B / (x - 1) ```

We multiply both sides by the denominator, (x 2)(x - 1), to clear the fractions:

``` 7x 18 A(x - 1) B(x 2) ```

Next, we expand and simplify the right-hand side:

``` 7x 18 Ax - A Bx 2B 7x 18 (A B)x (2B - A) ```

Now, we equate the coefficients of the corresponding terms on both sides:

For the coefficient of x:

A B 7

For the constant terms:

2B - A 18

Solving for A and B

We now solve the system of equations generated by the coefficients:

1. A B 7

2. 2B - A 18

Add equation 1 and equation 2 to eliminate A:

(A B) (2B - A) 7 18

3B 25

B 25 / 3

Substitute B back into equation 1:

A (25 / 3) 7

A 21 / 3 - 25 / 3

A -4 / 3

Partial Fractions

Therefore, the partial fraction decomposition of 7x 18 / (x^2 x - 3) is:

(-4/3) / (x 2) (25/3) / (x - 1)

This simplifies to:

``` (7x 18) / (x^2 x - 3) -(4/3) / (x 2) (25/3) / (x - 1) ```

Conclusion

Through the process of partial fraction decomposition, we have successfully broken down the complex rational expression 7x 18 / (x^2 x - 3) into simpler fractions. This method is particularly useful in solving higher-level algebraic problems and is a key skill in advanced mathematics.