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.) [(y^2 +2)^5(3y) - y^3(6)(y^2+2)^4(3y)] / [(y^2+2)^7]

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 to simplify the expression effectively.

Factoring

Factoring involves breaking down a polynomial into simpler components, or factors, that can be multiplied together to yield the original polynomial. This is essential in simplifying rational expressions, as it allows for the identification and cancellation of common factors in the numerator and denominator, streamlining the expression.

Common Factors

Common factors are elements that appear in both the numerator and denominator of a rational expression. Identifying and canceling these factors is a key step in simplification, as it reduces the expression to its simplest form. This process often involves recognizing patterns in polynomials, such as factoring out the greatest common factor (GCF).