Question

Find the partial fraction decomposition of each rational expression. $$ \frac{x^{2}+2 x+3}{\left(x^{2}+4\right)^{2}} $$

Short answer

\[\frac{x^{2}+2 x+3}{(x^{2}+4)^{2}} = \frac{x}{(x^{2}+4)} + \frac{2x - 1}{(x^{2}+4)^{2}}\]

Step-by-step solution

- Identify the Denominator

Notice that the denominator \((x^2 + 4)^2\) is already factored completely. This expression is a perfect square.

- Set up the Partial Fraction Decomposition Form

Given \((x^2 + 4)^2\), the general form for the partial fractions will include terms for \(\frac{A x + B}{x^2 + 4}\) and \(\frac{C x + D}{(x^2 + 4)^2}\). Hence, the partial fraction decomposition can be written as: \[\frac{x^{2}+2 x+3}{(x^{2}+4)^{2}} = \frac{A x + B}{x^{2}+4} + \frac{C x + D}{(x^{2}+4)^{2}}\]

- Clear the Denominator by Multiplying by \((x^2 + 4)^2\)

Multiply both sides by \((x^2 + 4)^2\) to eliminate the denominators: \[x^2 + 2x + 3 = (A x + B)(x^2 + 4) + (C x + D)\]

- Expand and Combine Like Terms

Expand the right-hand side and group the terms by powers of x. \[(A x + B)(x^2 + 4) = A x^3 + 4 A x + B x^2 + 4 B\] \Including \(C x + D\), we have: \[A x^3 + B x^2 + 4 A x + 4 B + C x + D = A x^3 + B x^2 + (4 A + C)x + (4 B + D)\]

- Compare Coefficients

Align the coefficients on both sides of the equation: \[\begin{aligned} &\text{Coefficient of } x^3: A = 0 \ &\text{Coefficient of } x^2: B = 1 \ &\text{Coefficient of } x: 4 A + C = 2 \ &\text{Constant term: } 4 B + D = 3 \end{aligned}\]

- Solve for Constants

From the coefficients we get:\(A = 0\)\(B = 1\)\ By substituting \(A = 0\) into the next equation:\(4(0) + C = 2 \Rightarrow C = 2\)\Then substituting \(B = 1\) into the last equation:\(4(1) + D = 3 \Rightarrow D = -1\)

- Write the Final Partial Fraction Decomposition

Substitute the constants back into the partial fraction form: \[\frac{x^{2}+2 x+3}{(x^{2}+4)^{2}} = \frac{x}{(x^{2}+4)} + \frac{2x - 1}{(x^{2}+4)^{2}}\]

Key concepts

Rational Expression

A rational expression is a fraction in which both the numerator and the denominator are polynomials. For example, \( \frac{x^2 + 2x + 3}{(x^2 + 4)^2} \) is a rational expression.

Rational expressions can sometimes be split into simpler partial fractions, making them easier to integrate or simplify.

In our example, we split \( \frac{x^2 + 2x + 3}{(x^2 + 4)^2} \) into simpler fractions like \( \frac{Ax + B}{x^2 + 4} \) and \( \frac{Cx + D}{(x^2 + 4)^2} \). This technique is known as partial fraction decomposition.

Rational expressions are often encountered in calculus and algebra, and understanding how to manipulate them is crucial for solving complex problems.

Factoring

Factoring is the process of decomposing a polynomial into a product of simpler polynomials. It simplifies complicated expressions and is an essential step in partial fraction decomposition.

For the denominator in our exercise, \( (x^2 + 4)^2 \) is already factored as it stands.

However, if we had a more complex denominator, we would first factor it completely. Factoring techniques include:

• Finding common factors
• Using special polynomial identities like difference of squares
• Quadratic factoring methods

Consistently practicing factoring will make working with rational expressions much more manageable.

Coefficients

Coefficients are the numerical or constant factors that multiply the variables in polynomial expressions. In the polynomial \( Ax + B \), A and B are coefficients.

Coefficients play a crucial role in aligning terms when solving equations, especially in the partial fraction decomposition process.

In the given problem, after clearing the denominator and expanding terms, we align coefficients of the same powers of x to form equations:
\[ \begin{aligned} &\text{Coefficient of } x^3: A = 0 \ &\text{Coefficient of } x^2: B = 1 \ &\text{Coefficient of } x: 4A + C = 2 \ &\text{Constant term: } 4B + D = 3 \end{aligned} \]

After setting up these equations, we solve for the unknown coefficients A, B, C, and D.

Polynomials

Polynomials are algebraic expressions consisting of variables and coefficients, combined using addition, subtraction, and multiplication, but not division. They are expressed in the form:
\[ a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \]

The degree of a polynomial is determined by the highest power of the variable. For example, in \( Ax^3 + Bx^2 + Cx + D \), the highest power is 3, making it a cubic polynomial.

Understanding polynomials is key to mastering many areas of mathematics. They are used in calculus, algebra, and geometry.

In our partial fraction decomposition example, both the numerator and denominator are polynomials. The process involves breaking down complex polynomial fractions into simpler parts. This simplification assists in further problem-solving steps, such as integration and differentiation.