Problem 41 - College Algebra

Question

Find the decomposition of the partial fraction for the irreducible non repeating quadratic factor. \(\frac{4 x^{2}+4 x+12}{8 x^{3}-27}\)

Step-by-Step Solution

Answer

The decomposition is \(\frac{2}{2x - 3} - \frac{4x + 2}{4x^2 + 6x + 9}\).

Step 1: Identify irreducible non-repeating quadratic factor

The denominator is given by \(8x^3 - 27\). This can be factored using the difference of cubes formula: \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\). For our denominator, \(a = 2x\) and \(b = 3\), so it becomes \((2x - 3)(4x^2 + 6x + 9)\). The term \((4x^2 + 6x + 9)\) is an irreducible quadratic.

Step 2: Set up the partial fraction decomposition

Since the denominator factors as \((2x - 3)(4x^2 + 6x + 9)\), we can decompose the fraction into two parts: \(\frac{A}{2x - 3} + \frac{Bx + C}{4x^2 + 6x + 9}\), where \(A\), \(B\), and \(C\) are constants to be determined.

Step 3: Clear the denominators

Multiply each term by the common denominator \((2x - 3)(4x^2 + 6x + 9)\) to eliminate the fractions: \[4x^2 + 4x + 12 = A(4x^2 + 6x + 9) + (Bx + C)(2x - 3)\].

Step 4: Expand and simplify the equation

Distribute on the right: \(A(4x^2 + 6x + 9) = 4Ax^2 + 6Ax + 9A\) and \((Bx + C)(2x - 3) = 2Bx^2 + (2C - 3B)x - 3C\). Combine these to form: \[4x^2 + 4x + 12 = (4A + 2B)x^2 + (6A + 2C - 3B)x + (9A - 3C)\].

Step 5: Equate coefficients

From the equation \(4x^2 + 4x + 12 = (4A + 2B)x^2 + (6A + 2C - 3B)x + (9A - 3C)\), match coefficients: \(4A + 2B = 4\), \(6A + 2C - 3B = 4\), and \(9A - 3C = 12\).

Step 6: Solve the system of equations

Solve \(4A + 2B = 4\), which simplifies to \(2A + B = 2\). \(6A + 2C - 3B = 4\) and \(9A - 3C = 12\) form another system. From \(9A - 3C = 12\), solve for \(C = 3A - 4\). Substitute back to find \(A = 2\), \(B = -4\), and \(C = 2\).

Step 7: Write the partial fraction decomposition

Using the values obtained, the partial fraction decomposition is \(\frac{2}{2x - 3} - \frac{4x + 2}{4x^2 + 6x + 9}\).

Key Concepts

Irreducible Quadratic Factor

To decompose an algebraic expression into partial fractions, it's important to understand the concept of an **irreducible quadratic factor**. An irreducible quadratic factor is a polynomial of the form \(ax^2 + bx + c\) that cannot be factored into linear terms using real numbers. This means, it has no real roots; its discriminant \(b^2 - 4ac < 0\). In the given exercise, we identified \(4x^2 + 6x + 9\) as the irreducible quadratic factor. It cannot be factored further using real numbers, making it irreducible. Recognizing such quadratics helps in setting up the correct partial fraction decomposition.

Difference of Cubes Formula

The **difference of cubes formula** is a handy tool in factorizing expressions of the form \(a^3 - b^3\). This formula states that \(a^3 - b^3\) can be expressed as \((a-b)(a^2 + ab + b^2)\). It simplifies seemingly complex expressions into manageable parts.

• In our exercise, the denominator \(8x^3 - 27\) is a difference of cubes, since \(8x^3\) is \((2x)^3\) and \(27\) is \(3^3\).
• Applying the formula, we factored it as \((2x - 3)(4x^2 + 6x + 9)\). This step was crucial in simplifying the expression and determining the irreducible quadratic factor in this context.

By utilizing such factoring techniques, complex fractions become easier to decompose.

System of Equations

In partial fraction decomposition, solving a **system of equations** is often required to find the constants of the decomposed fractions. In our solution, setting up the correct equations involved equating coefficients from a polynomial identity, which then allows us to solve for unknown constants.

• Given equations were \(4A + 2B = 4\), \(6A + 2C - 3B = 4\), and \(9A - 3C = 12\).
• Solving these equations systematically by substitution and elimination allows us to find specific values for constants \(A\), \(B\), and \(C\).

Approaching such systems involves logical deduction and basic algebraic manipulation, essential skills in mathematics that emphasize relationships among variables.

Algebraic Expressions

**Algebraic expressions** are the building blocks in mathematics consisting of variables, constants, and functions combined using mathematical operations like addition and multiplication. In the context of partial fraction decomposition:

• The numerator \(4x^2 + 4x + 12\) and the denominator \(8x^3 - 27\) are composed of algebraic expressions.
• The factorization into \((2x - 3)(4x^2 + 6x + 9)\) displays each component's structure.
• Balancing and rearranging terms to match coefficients involves understanding how to manipulate these expressions.

Mastering algebraic expressions is crucial; it aids in recognizing patterns, simplifying, and solving complex expressions in mathematics.