Answer each question. By what expression should we multiply each side of (3x - 1)/(x(2x^2 + 1)^2) = A/x + (Bx + C)/(2x^2 + 1) + (Dx + E)/(2x^2 + 1)^2 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 left side has \(x(2x^2 + 1)^2\), and the right side has denominators \(x\), \$2x^2 + 1\(, and \)(2x^2 + 1)^2$.
2. To eliminate all fractions, multiply both sides of the equation by the least common denominator (LCD) of all these denominators.
3. The LCD must include each factor to the highest power it appears in any denominator, so the LCD is \(x(2x^2 + 1)^2\).
4. Multiply each term on both sides of the equation by \(x(2x^2 + 1)^2\) to clear all denominators.
5. After multiplication, the equation will be free of fractions, allowing you to equate the numerators and solve for the constants \(A\), \(B\), \(C\), \(D\), and \(E\).

Key Concepts

Partial Fraction Decomposition

Partial fraction decomposition breaks a complex rational expression into simpler fractions whose denominators are factors of the original denominator. This technique is useful for integration and solving equations involving rational expressions. Understanding the form of the decomposition helps identify the denominators involved.

Common Denominator and Clearing Fractions

To eliminate fractions in an equation, multiply both sides by the least common denominator (LCD) of all fractional terms. The LCD is the smallest expression that contains all denominators as factors. Multiplying by the LCD clears denominators, resulting in a polynomial equation easier to solve.

Factoring and Identifying Denominators

Recognizing and factoring denominators is essential to find the LCD. In this problem, denominators include x, (2x^2 + 1), and (2x^2 + 1)^2. The LCD must include each factor at its highest power to clear all fractions effectively.