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.) [(x² +1)⁴(2x) - x²(4)(x²+1)³(2x)] / [(x²+1)⁸]

Verified step by step guidance

1. Start by writing the given expression clearly: \(\frac{(x^2 + 1)^4 (2x) - x^2 (4) (x^2 + 1)^3 (2x)}{(x^2 + 1)^8}\).
2. Look for common factors in the numerator. Notice that both terms contain \((x^2 + 1)^3\) and \$2x\(. Factor these out: \)2x (x^2 + 1)^3 \left[(x^2 + 1) - 4x^2\right]$.
3. Simplify the expression inside the brackets: \((x^2 + 1) - 4x^2 = 1 - 3x^2\).
4. Rewrite the numerator as \$2x (x^2 + 1)^3 (1 - 3x^2)\( and keep the denominator as \)(x^2 + 1)^8$.
5. Divide out the common factor \((x^2 + 1)^3\) from numerator and denominator, which leaves \(\frac{2x (1 - 3x^2)}{(x^2 + 1)^5}\) as the simplified expression.

Key Concepts

Factoring Polynomial Expressions

Factoring involves rewriting polynomial expressions as products of simpler polynomials. It helps to identify common factors in the numerator and denominator, which can be canceled to simplify rational expressions. Recognizing patterns like the distributive property or common terms is essential.

Properties of Exponents

Understanding how to manipulate exponents is crucial when simplifying expressions with powers, such as (x^2 + 1)^n. Key rules include multiplying powers when bases are the same and subtracting exponents when dividing like bases. This allows simplification of terms involving powers efficiently.

Simplifying Rational Expressions

A rational expression is a fraction where numerator and denominator are polynomials. Simplifying involves factoring both parts, canceling common factors, and reducing the expression to its simplest form. Assuming variables represent positive real numbers ensures no issues with domain restrictions during simplification.