Simplify each rational expression. Assume all variable expressions represent positive real numbers. (Hint: Use factoring and divide out any common factors as a first step.) [(x^2 +1)^4(2x) - x^2(4)(x^2+1)^3(2x)] / [(x^2+1)^8]

Key Concepts

Rational Expressions

Rational expressions are fractions where the numerator and denominator are polynomials. Understanding how to manipulate these expressions is crucial for simplification. This includes recognizing common factors and applying algebraic operations such as addition, subtraction, multiplication, and division. In this context, simplifying involves factoring and reducing the expression to its simplest form.

Factoring

Factoring is the process of breaking down a polynomial into its constituent factors, which can be multiplied together to yield the original polynomial. This is essential for simplifying rational expressions, as it allows for the identification and cancellation of common factors in the numerator and denominator. Mastery of factoring techniques, such as grouping and using special products, is vital for effective simplification.

Common Factors

Common factors are terms that appear in both the numerator and denominator of a rational expression. Identifying and canceling these factors is a key step in simplifying rational expressions. This process not only reduces the complexity of the expression but also helps in understanding the relationships between the variables involved. Recognizing common factors is fundamental to achieving a simplified form of the expression.