Answer each question. By what expression should we multiply each side of (3x - 2)/(x + 4)(3x^2 + 1) = A/(x + 4) + (Bx + C)/(3x^2 + 1) so that there are no fractions in the equation?

Verified step by step guidance

1. Identify the denominators on both sides of the equation. The denominators are \((x + 4)\) and \((3x^2 + 1)\).
2. To eliminate the fractions, multiply each term on both sides of the equation by the least common denominator (LCD) of all denominators present.
3. The LCD is the product of the distinct factors in the denominators, which is \((x + 4)(3x^2 + 1)\).
4. Multiply every term on both sides of the equation by this LCD: \((x + 4)(3x^2 + 1)\).
5. This multiplication will clear all denominators, resulting in an equation without fractions, making it easier to solve for \(A\), \(B\), and \(C\).

Key Concepts

Partial Fraction Decomposition

Partial fraction decomposition is a method used to express a rational function as a sum of simpler fractions. It involves breaking down a complex fraction into terms with simpler denominators, often linear or quadratic factors, to facilitate integration or solving equations.

Common Denominator Multiplication

To eliminate fractions in an equation, multiply both sides by the least common denominator (LCD) of all fractional terms. This clears denominators, resulting in a polynomial equation that is easier to solve or manipulate.

Factoring and Identifying Denominators

Understanding how to factor expressions and identify the denominators involved is crucial. In this problem, recognizing the denominators (x + 4) and (3x^2 + 1) helps determine the LCD, which is their product, to clear fractions effectively.