Simplify each complex fraction. [ 2/[(x+h)² + 16] - 2/(x²+16)] / h

Verified step by step guidance

1. Identify the complex fraction: the numerator is the difference of two fractions, and the denominator is a single variable \(h\). The expression is: \(\frac{\frac{2}{(x+h)^2 + 16} - \frac{2}{x^2 + 16}}{h}\).
2. Focus on simplifying the numerator first, which is \(\frac{2}{(x+h)^2 + 16} - \frac{2}{x^2 + 16}\). To combine these two fractions, find a common denominator, which is \(\left((x+h)^2 + 16\right) \left(x^2 + 16\right)\).
3. Rewrite each fraction with the common denominator: \(\frac{2(x^2 + 16)}{\left((x+h)^2 + 16\right) \left(x^2 + 16\right)} - \frac{2((x+h)^2 + 16)}{\left((x+h)^2 + 16\right) \left(x^2 + 16\right)}\).
4. Combine the numerators over the common denominator: \(\frac{2(x^2 + 16) - 2((x+h)^2 + 16)}{\left((x+h)^2 + 16\right) \left(x^2 + 16\right)}\).
5. Now, rewrite the entire original expression as a single fraction by dividing this result by \(h\), which is equivalent to multiplying by \(\frac{1}{h}\). So the expression becomes: \(\frac{2(x^2 + 16) - 2((x+h)^2 + 16)}{h \cdot \left((x+h)^2 + 16\right) \left(x^2 + 16\right)}\).

Key Concepts

Complex Fractions

A complex fraction is a fraction where the numerator, denominator, or both contain fractions themselves. Simplifying involves rewriting the expression as a single fraction by finding common denominators or multiplying by the least common denominator to eliminate the smaller fractions.

Common Denominator

To combine or subtract fractions, you must express them with a common denominator. This involves finding the least common multiple of the denominators and rewriting each fraction accordingly, allowing the numerators to be combined directly.

Difference Quotient and Simplification

The given expression resembles a difference quotient, often used in calculus. Simplifying such expressions requires careful algebraic manipulation, including expanding binomials, combining like terms, and factoring to reduce the expression, especially when dividing by a variable like h.