Question

Find the partial fraction decomposition of each rational expression. $$ \frac{x^{2}+2 x+3}{(x+1)\left(x^{2}+2 x+4\right)} $$

Short answer

\( \frac{1}{x+1} + \frac{-1}{x^2+2x+4} \)

Step-by-step solution

- Set Up the Partial Fractions

Express the given rational expression as a sum of partial fractions. Identify that \(\frac{x^2 + 2x + 3}{(x+1)(x^2+2x+4)}\) can be decomposed into \(\frac{A}{x+1} + \frac{Bx+C}{x^2+2x+4} \).

- Multiply by the Common Denominator

Multiply both sides by the common denominator \( (x+1)(x^2+2x+4) \) to eliminate the denominators: \( x^2 + 2x + 3 = A(x^2 + 2x + 4) + (Bx + C)(x + 1) \).

- Expand and Combine Like Terms

Expand the equation: \( x^2 + 2x + 3 = A(x^2 + 2x + 4) + (Bx^2 + Bx + Cx + C) \). Combine like terms to get: \( x^2 + 2x + 3 = Ax^2 + 2Ax + 4A + Bx^2 + Bx + Cx + C \). Simplify to: \( x^2 + 2x + 3 = (A + B)x^2 + (2A + B + C)x + (4A + C) \).

- Set Up the System of Equations

Set up equations by equating coefficients from both sides: \( 1. A + B = 1 \) \( 2. 2A + B + C = 2 \) \( 3. 4A + C = 3 \).

- Solve the System of Equations

Solve the system of equations: \( A + B = 1 \), \( 2A + B + C = 2 \), \( 4A + C = 3 \). From equation 3, \( C = 3 - 4A \). Substitute C into equation 2: \( 2A + B + (3 - 4A) = 2 \), simplifies to \( -2A + B + 3 = 2 \), \( -2A + B = -1 \). Now solve simultaneously with \( A + B = 1 \) to find \( A = 1 \), \( B = 0 \), \( C = -1 \).

- Write the Partial Fraction Decomposition

Substitute the values of \( A \), \( B \), and \( C \) back into the partial fractions to get: \[ \frac{x^2+2x+3}{(x+1)(x^2+2x+4)} = \frac{1}{x+1} + \frac{-1}{x^2+2x+4} \].

Key concepts

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. They can look complex, but with careful handling, they become manageable. To simplify or decompose these expressions, we often turn them into a sum of simpler fractions. This is what we call partial fraction decomposition.
Partial fraction decomposition is especially useful when integrating rational expressions or solving differential equations.
In this exercise, we're dealing with the rational expression \(\frac{x^{2}+2 x+3}{(x+1)(x^{2}+2 x+4)}\). We'll break it down into simpler parts by expressing it as a sum of fractions.

System of Equations

A system of equations is a set of equations with the same variables. Solving the system means finding values for the variables that satisfy all the equations simultaneously.
In partial fraction decomposition, we often encounter a system of equations when matching coefficients from the expanded form.
For our example, we ended up with the system:

• \(A + B = 1\)
• \(2A + B + C = 2\)
• \(4A + C = 3\)

Solving this system helps us find the values of our constants (A, B, and C). It often requires substituting and rearranging equations, which is a core skill in algebra.

Algebra

Algebra is the branch of mathematics dealing with symbols and the rules for manipulating those symbols. It is essential for solving equations and understanding the relationships between variables.
In the partial fraction decomposition problem, algebra helps us rearrange and solve the equations. We start by expressing our rational expression as a sum of simpler fractions. Then, we use algebra to find the values of the constants in those fractions.
The crucial steps involve:

• Expanding the equations after multiplying out the common denominators
• Combining like terms
• Setting up and solving the resulting system of equations

The algebra ensures that each step is logically consistent and leads to the correct values for the constants, simplifying the expression efficiently and accurately.