Problem 22 - Precalculus: Functions and Graphs

Question

Find the partial fraction decomposition. \(\frac{3 x^{3}+13 x-1}{\left(x^{2}+4\right)^{2}}\)

Step-by-Step Solution

Answer

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

Step 1: Identify the form of partial fractions

The denominator, \((x^2 + 4)^2\), is a repeated irreducible quadratic factor. Therefore, the decomposition is \(\frac{3x^3 + 13x - 1}{(x^2 + 4)^2} = \frac{Ax + B}{x^2 + 4} + \frac{Cx + D}{(x^2 + 4)^2}\).

Step 2: Rewrite the equation with common denominator

Combine the right-hand side under a single denominator: \(\frac{Ax + B}{x^2 + 4} + \frac{Cx + D}{(x^2 + 4)^2} = \frac{(Ax + B)(x^2 + 4) + (Cx + D)}{(x^2 + 4)^2}\) and equate it to the original fraction.

Step 3: Expand and simplify the numerator

Expand \((Ax + B)(x^2 + 4)\) to get \(Ax^3 + Ax \cdot 4 + Bx^2 + 4B = Ax^3 + 4Ax + Bx^2 + 4B\). So, \(Ax^3 + Bx^2 + (4A) x + 4B + Cx + D\).

Step 4: Equate coefficients from the numerators

Equate the coefficients from the expanded numerator: \(A = 3\) for \(x^3\), \(B = 0\) for \(x^2\), \(4A + C = 13\) for \(x\), and \(4B + D = -1\) for the constant term.

Step 5: Solve for constants

Using \(A = 3\) and \(B = 0\), the equations become \(4 \cdot 3 + C = 13\) leading to \(C = 1\) and \(4 \cdot 0 + D = -1\) leading to \(D = -1\).

Step 6: Write the final decomposition

Insert these constants back into the partial fraction form: \(\frac{3x + 0}{x^2 + 4} + \frac{x - 1}{(x^2 + 4)^2}\). This simplifies to \(\frac{3x}{x^2 + 4} + \frac{x - 1}{(x^2 + 4)^2}\).

Key Concepts

Repeated Irreducible Quadratic Factor

In partial fraction decomposition, a repeated irreducible quadratic factor refers to quadratic expressions in the denominator that cannot be further simplified or factored into linear terms over the real numbers. Consider the expression \((x^2 + 4)^2\). Here, \(x^2 + 4\) is irreducible over the reals because it cannot be broken down into simpler real factors. The repetition is signified by its exponent, 2. This affects the decomposition format, requiring us to account for each power separately.

For such repeated factors, the decomposition consists of terms for both the factor and its powers up to the highest repeat. For example,

• \(\frac{Ax + B}{x^2 + 4}\) handles the single power.
• \(\frac{Cx + D}{(x^2 + 4)^2}\) covers the repeated power.

This ensures that we represent the full spectrum of possible solutions.

Numerator Expansion

Once the decomposition form is set, our next step is to combine terms over a common denominator. This involves expanding the numerators of the fraction expressions. Take the decomposition \(\frac{Ax + B}{x^2 + 4} + \frac{Cx + D}{(x^2 + 4)^2}\). We need to express this as a single fraction:

• First, expand \((Ax + B)(x^2 + 4)\) to match the original context.
• This results in terms \(Ax^3 + 4Ax + Bx^2 + 4B\), which needs meticulous calculation.

Afterwards, add the term \((Cx + D)\), to ensure that the expression is fully over \((x^2 + 4)^2\). This approach allows one to prepare for the next crucial phase of comparing coefficients.

Coefficient Comparison

Once the numerators are expanded, the polynomial needs to be equated to the original fraction for coefficient comparison. Each coefficient from the original polynomial \(3x^3 + 13x - 1\) must match the respective expanded terms:

• For \(x^3\), set \(A = 3\).
• For \(x^2\), have \(B = 0\).
• Balance the \(x\) terms, giving the equation \(4A + C = 13\).
• Lastly, align the constant terms, requiring \(4B + D = -1\).

Solving these equations helps to determine unknown constants in the numerators. This is important as it transforms algebraic manipulation into understandable variables with specific values.

Quadratic Denominator

Partial fraction decomposition often encounters quadratic denominators which can either be irreducible or reducible. Quadratic denominators like \(x^2 + 4\) highlight how different types of expressions influence the decomposition process. When dealing with irreducible quadratic factors, the assignation in the decomposition should include linear expressions in the numerator, \((Ax + B)\), to cover potential variations.

Managing quadratics involves:

• Recognizing how they fail to factor over real numbers.
• Ensuring appropriate numerators are selected, reflecting potential solutions omitted otherwise.
• Providing a format that can capture the behavior of the original rational expression.

Quadratic denominators add a layer of complexity but help to extend understanding by considering advanced algebraic inputs during decomposition.