Problem 46 - Calculus Single Variable

Question

What happens if you attempt a partial fraction decomposition of \(1 /\left(x^{2}+3\right)^{4}\) into $$ \frac{A_{1} x+B_{1}}{x^{2}+3}+\frac{A_{2} x+B_{2}}{\left(x^{2}+3\right)^{2}}+\frac{A_{3} x+B_{3}}{\left(x^{2}+3\right)^{3}}+\frac{A_{4} x+B_{4}}{\left(x^{2}+3\right)^{4}} ? $$ (Brute force calculation should be a last resort.)

Step-by-Step Solution

Answer

The partial fraction decomposition is not straightforward due to complex dependencies; it directly reduces to standard-solving conditions or symmetries rather than clear constant values.

Step 1: Understand the Denominator

The denominator of the expression is \((x^2 + 3)^4\). It is a power of a quadratic expression \(x^2 + 3\), raised to the 4th power.

Step 2: Setup Partial Fraction Decomposition

According to the problem, the decomposition is set up as follows: \[ \frac{1}{(x^2 + 3)^4} = \frac{A_1 x + B_1}{x^2 + 3} + \frac{A_2 x + B_2}{(x^2 + 3)^2} + \frac{A_3 x + B_3}{(x^2 + 3)^3} + \frac{A_4 x + B_4}{(x^2 + 3)^4} \]This means expressing \(1\) over the complex fraction as a sum of fractions, each with a linear numerator over increasing powers of the base quadratic \(x^2 + 3\).

Step 3: Combine the Right-Hand Side

Start by writing the right-hand side fractions over a common denominator \((x^2 + 3)^4\):\[ \frac{(A_1 x + B_1)(x^2 + 3)^3 + (A_2 x + B_2)(x^2 + 3)^2 + (A_3 x + B_3)(x^2 + 3) + (A_4 x + B_4)}{(x^2 + 3)^4} \] This should equal \(\frac{1}{(x^2 + 3)^4}\).

Step 4: Equate Numerators

The numerators must be equal:\[ (A_1 x + B_1)(x^2 + 3)^3 + (A_2 x + B_2)(x^2 + 3)^2 + (A_3 x + B_3)(x^2 + 3) + (A_4 x + B_4) = 1 \]This equation shows the combination of all terms to produce exactly \(1\).

Step 5: Solving the System of Equations

Expand and simplify the left-hand side. Each coefficient from the expanded polynomial must match the corresponding coefficient of \(x\) in \(1\). However, because there is only a single constant term on the right (1) and no powers of \(x\), many coefficients on the left must be zero due to matching terms by degree.Solve for \(A_1, B_1, A_2, B_2, A_3, B_3, A_4, \) and \(B_4\). Given symmetry and the requirement to sum to reach only the constant "1," each term's contribution must incidentally produce the same effect as a direct subtraction, not leaving meaningful assignments for constants \(A_1, A_2, A_3, A_4, B_1, B_2, B_3, B_4\) beyond the last term approximating methods.

Key Concepts

Quadratic Expression

A quadratic expression is a polynomial of degree two, typically in the form \(ax^2 + bx + c\). Understanding how it behaves when performing algebraic manipulations is key to mastering topics like partial fraction decomposition. In this specific problem, our quadratic expression is \(x^2 + 3\). It is quite simple since there is no \(x\) term, and the constant is 3, indicating a parabola that opens upwards with a vertex at the origin shifted vertically.

In the context of our exercise, this quadratic expression forms the base of a composite polynomial \((x^2 + 3)^4\). The task is to decompose a reciprocal of this polynomial using partial fractions, a common technique in calculus, especially within integration. Pay special attention when the base quadratic expression is raised to higher powers, as this will change the number and form of fractions in the decomposition.

Algebraic Manipulation

Algebraic manipulation involves the use of operations to simplify and transform expressions. In the process of solving mathematical problems, algebraic manipulation can be crucial for rewriting expressions in a more workable form.

In our decomposition exercise, algebraic manipulation comes into play when setting up the partial fraction decomposition and solving for unknown coefficients. After setting up the decomposition with terms such as \(\frac{A_i x + B_i}{(x^2 + 3)^i}\), you will combine these fractions over a single denominator, aligning them with the left side of the equation. The challenging part involves expanding terms with powers and coefficients, ensuring each part contributes correctly to match a simpler desired form (like "1" in this scenario). Often, this requires keen algebra skills to correctly expand polynomials and simplify resulting expressions.

When matching terms by degree, a critical step is rearranging and collecting like terms. This allows you to equate the two sides successfully, distinguishing constants from variables to solve the equation effectively in the context of a "system of equations."

System of Equations

A system of equations consists of multiple equations that are solved simultaneously. In the context of our exercise on partial fraction decomposition, solving a system of equations is the final step where you determine the values of unknown coefficients.

The initial step involves equating the numerators of the right and left-hand sides. This creates a complex polynomial equation with terms grouped by powers of \(x\). The right-hand side will only have a constant term due to the nature of this specific problem, forcing all \(x\)-related terms on the left to equal zero. This requirement transforms the problem into solving a system where coefficients of each power of \(x\) must match.

Through algebraic manipulation, the problem reduces to simpler linear equations. Solve these equations to find the coefficients \(A_1, B_1, A_2, B_2, A_3, B_3, A_4,\) and \(B_4\). It is crucial to recognize that symmetry and balance play a role in how the coefficients relate, often simplifying potentially complex interactions. Using these techniques will help you find consistent values for the coefficients that solve the original decomposition problem.