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 least common denominator (LCD) of all the fractions in the equation.
2. The denominators are: x, (2x^2 + 1), and (2x^2 + 1)^2.
3. The LCD is x(2x^2 + 1)^2 because it is the highest power of each factor present in the denominators.
4. Multiply each term in the equation by the LCD, x(2x^2 + 1)^2, to eliminate the fractions.
5. Simplify each term after multiplication to ensure all fractions are cleared.

Key Concepts

Rational Expressions

Rational expressions are fractions where the numerator and denominator are polynomials. Understanding how to manipulate these expressions is crucial for solving equations involving them. In this context, we need to eliminate fractions to simplify the equation, which often involves finding a common denominator.

Common Denominator

A common denominator is a shared multiple of the denominators of two or more fractions. To eliminate fractions in an equation, we multiply through by the least common denominator (LCD) of all terms. This process allows us to clear the fractions and work with a polynomial equation instead.

Polynomial Equations

Polynomial equations are mathematical statements that set a polynomial expression equal to another expression. Solving these equations often involves factoring, expanding, or applying the quadratic formula. Once fractions are eliminated, the resulting polynomial equation can be manipulated using algebraic techniques to find the values of the variables.