Problem 45 - Precalculus

Question

Find the partial-fraction decomposition of \(\frac{3 x^{2}+3 x+1}{x\left(x^{2}+1\right)}\) Solution: Write the partial-fraction decomposition form. $$\frac{3 x^{2}+3 x+1}{x\left(x^{2}+1\right)}=\frac{A}{x}+\frac{B}{x^{2}+1}$$ Multiply both sides by the \(\operatorname{LCD} x\left(x^{2}+1\right)\) Eliminate the parentheses. $$\begin{aligned}&3 x^{2}+3 x+1=A\left(x^{2}+1\right)+B x\\\&3 x^{2}+3 x+1=A x^{2}+B x+A\end{aligned}$$ Matching like terms leads to three equations. $$A=3, B=3, \text { and } A=1$$ This is incorrect. What mistake was made?

Step-by-Step Solution

Answer

The mistake was using \( \frac{B}{x^2+1} \) instead of \( \frac{Bx+C}{x^2+1} \). Proper decomposition is \( \frac{1}{x} + \frac{2x+3}{x^2+1} \).

Step 1: Understand the Initial Equation

The given equation for partial fraction decomposition is \( \frac{3x^2 + 3x + 1}{x(x^2 + 1)} \). This expression suggests a decomposition form involving linear terms in the denominator: \( \frac{A}{x} + \frac{Bx + C}{x^2 + 1} \) because \( x^2 + 1 \) is an irreducible quadratic factor.

Step 2: Write the Correct Decomposition Form

The appropriate partial-fraction decomposition form should be \( \frac{3x^2 + 3x + 1}{x(x^2 + 1)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} \). This accounts for the linear term and the irreducible quadratic term in the denominator.

Step 3: Eliminate Denominator by Multiplying LCD

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

Step 4: Simplify the Right Side

Expand the right side: \( A(x^2 + 1) + (Bx + C)x = Ax^2 + A + Bx^2 + Cx \). Combine like terms to get \( (A + B)x^2 + Cx + A \).

Step 5: Equate and Get System of Equations

By equating coefficients from \( 3x^2 + 3x + 1 = (A + B)x^2 + Cx + A \), we get a system: \( A + B = 3 \), \( C = 3 \), \( A = 1 \).

Step 6: Solve the System of Equations

From \( A = 1 \), substitute into \( A + B = 3 \) to find \( B = 3 - 1 = 2 \). With \( C = 3 \), our coefficients are determined.

Step 7: Final Partial Fraction Decomposition

The correct partial-fraction decomposition of \( \frac{3x^2 + 3x + 1}{x(x^2 + 1)} \) is \( \frac{1}{x} + \frac{2x + 3}{x^2 + 1} \).

Key Concepts

Rational Expressions

A rational expression is essentially a fraction where both the numerator and the denominator are polynomials. In partial fraction decomposition, we start with a rational expression—just like \(\frac{3x^2 + 3x + 1}{x(x^2 + 1)}\)—and aim to express it as a sum of simpler rational expressions. This helps because it's often easier to integrate or differentiate these simpler expressions, or solve other algebraic problems.
To decompose a rational expression, our first step is examining its denominator. Understanding and categorizing the factors in the denominator, such as whether they are linear or quadratic, is crucial. This leads to determining the proper form of partial fractions that will be used, which brings us to irreducible quadratic factors.

Irreducible Quadratic Factor

An irreducible quadratic factor is a term in the denominator that cannot be factored into linear terms using real numbers. For instance, in our given expression, \(x^2 + 1\) is an irreducible quadratic. It's irreducible because it cannot be further simplified to linear terms unless complex numbers are considered.
When dealing with irreducible quadratics in partial fraction decomposition, it's essential to account for both a linear and a constant term in the numerator. This is why, in the correct decomposition form of our exercise, we use \( \frac{Bx + C}{x^2 + 1} \). The presence of both \(B\) and \(C\) as coefficients allows us to accurately equate and solve the coefficients necessary for complete decomposition. Understanding these components helps set up a system of equations to solve for \(A, B, \) and \(C\) as necessary.

System of Equations

Using a system of equations is a fundamental step in finding the coefficients for partial fraction decomposition. Once the correct decomposition structure is set, each distinct term is expanded and like terms are aggregated, enabling us to compare coefficients with the original polynomial in the numerator.
For the expression \(3x^2 + 3x + 1\), after expanding and rearranging our guessed decomposition form (\(\frac{1}{x} + \frac{2x + 3}{x^2 + 1}\)), we equate the coefficients of \(x^2, x,\) and the constant term. This results in a system of equations:

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

These equations can then be solved to find \(A, B, \) and \(C\). For example, substituting \(A = 1\) into the first equation gives \(B = 2\). The result is all coefficients identified properly, finalizing the correct form. By mastering this step, students ensure their decompositions are correctly aligned with the given expression.