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.) [4(x²- 1)³ + 8x(x²-1)⁴] / [16(x²-1)³]

Verified step by step guidance

1. Start by examining the given rational expression: \(\frac{4(x^2 - 1)^3 + 8x(x^2 - 1)^4}{16(x^2 - 1)^3}\). Notice that both the numerator and denominator contain powers of the expression \((x^2 - 1)\).
2. Factor out the greatest common factor (GCF) from the numerator. The GCF is \$4(x^2 - 1)^3\(, so rewrite the numerator as \)4(x^2 - 1)^3 \left[1 + 2x(x^2 - 1)\right]$.
3. Rewrite the entire expression using the factored numerator: \(\frac{4(x^2 - 1)^3 \left[1 + 2x(x^2 - 1)\right]}{16(x^2 - 1)^3}\).
4. Cancel out the common factors in the numerator and denominator. Since both have \$4(x^2 - 1)^3\(, divide numerator and denominator by this term to simplify the expression to \)\frac{1 + 2x(x^2 - 1)}{4}$.
5. Finally, expand the term \$2x(x^2 - 1)\( in the numerator to get \)2x^3 - 2x\(, then write the simplified expression as \)\frac{1 + 2x^3 - 2x}{4}$. This is the simplified form of the original rational expression.

Key Concepts

Factoring Polynomials

Factoring involves rewriting a polynomial as a product of simpler polynomials or expressions. Recognizing common patterns like difference of squares or factoring out the greatest common factor helps simplify expressions and is essential before performing operations like division.

Simplifying Rational Expressions

A rational expression is a fraction where the numerator and denominator are polynomials. Simplifying involves factoring both parts and canceling out common factors, reducing the expression to its simplest form while considering domain restrictions.

Properties of Exponents

Understanding how to manipulate expressions with exponents, such as multiplying powers with the same base or factoring out common powers, is crucial. This helps in combining like terms and simplifying expressions involving powers efficiently.