Introduction to Partial Fractions

Partial fractions are a technique used in mathematics to decompose a rational function into simpler fractions. This process is particularly useful for integration and simplification of complex algebraic expressions. In this article, we will explore how to express a specific rational function, 2x^2 x - 3 / 9 - x^2, in partial fractions.

Step 1: Factor the Denominator

The first step in expressing a rational function in partial fractions is to factor the denominator. For the function 2x^2 x - 3 / 9 - x^2, we start by factoring the denominator 9 - x^2, which can be rewritten using the difference of squares formula:

9 - x^2 (3 - x)(3 x)

Step 2: Set Up the Partial Fraction Decomposition

After factoring the denominator, we set up the partial fraction decomposition as follows:

(2x^2 x - 3) / ((3 - x)(3 x)) A / (3 - x) B / (3 x)

Here, A and B are constants that we need to determine.

Step 3: Multiply Through by the Denominator

Next, we multiply both sides of the equation by the denominator to eliminate the fractions:

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

Step 4: Expand the Right Side

Expanding the right-hand side of the equation gives us:

2x^2 x - 3 A(3 x) B(3 - x) 3A Ax 3B - Bx

Simplifying this expression, we get:

2x^2 x - 3 3A - Bx Ax 3B

Step 5: Set Up the System of Equations

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

Coefficient of x^2: 0 2 (no x^2 term on the right side) Coefficient of x: 1 A - B Constant term: -3 3A 3B

From the first equation, since we have no x^2 term on the right side, we can infer that the x^2 term is already satisfied.

Step 6: Solve the System

From the second equation (1 A - B), we can express A in terms of B:

A 1 B

Substituting this into the third equation (3A 3B -3), we get:

3(1 B) 3B -3 3 3B 3B -3 6B -6 B -1

Substituting B -1 back into the equation A 1 B, we get:

A 1 - 1 0

So, the partial fraction decomposition is:

(2x^2 x - 3) / ((3 - x)(3 x)) 0 / (3 - x) - 1 / (3 x) -1 / (3 x)

Step 7: Write the Final Expression

The partial fraction decomposition of the given rational function is:

(2x^2 x - 3) / (9 - x^2) -1 / (3 x)

Conclusion

In this article, we have demonstrated how to express the rational function (2x^2 x - 3) / (9 - x^2) in partial fractions. The final expression is -1 / (3 x).

Related Keywords

Rational Functions, Partial Fractions, Algebraic Expressions