Simplify each complex fraction. $\frac{\frac{3}{p^2 - 16} + p}{\frac{1}{p - 4}}$

Verified step by step guidance

1. Identify the complex fraction: \(\frac{\frac{3}{p^{2} - 16} + p}{\frac{1}{p - 4}}\).
2. Recognize that \(p^{2} - 16\) is a difference of squares and factor it as \(p^{2} - 16 = (p - 4)(p + 4)\).
3. Rewrite the numerator by expressing \(p\) as a fraction with denominator \((p - 4)(p + 4)\) to combine the terms: \(\frac{3}{(p - 4)(p + 4)} + \frac{p(p - 4)(p + 4)}{(p - 4)(p + 4)}\).
4. Combine the fractions in the numerator over the common denominator \((p - 4)(p + 4)\): \(\frac{3 + p(p - 4)(p + 4)}{(p - 4)(p + 4)}\).
5. Divide the combined numerator by the denominator \(\frac{1}{p - 4}\) by multiplying the numerator by the reciprocal of the denominator: \(\frac{3 + p(p - 4)(p + 4)}{(p - 4)(p + 4)} \times (p - 4)\), then simplify the expression.

Key Concepts

Complex Fractions

A complex fraction is a fraction where the numerator, denominator, or both contain fractions themselves. Simplifying involves rewriting the expression to eliminate the smaller fractions, often by finding a common denominator or multiplying numerator and denominator by the least common denominator (LCD).

Factoring Polynomials

Factoring involves expressing a polynomial as a product of simpler polynomials. Recognizing special forms like the difference of squares, e.g., p^2 - 16 = (p - 4)(p + 4), helps simplify expressions and cancel common factors in fractions.

Operations with Rational Expressions

Rational expressions are fractions with polynomials in numerator and denominator. Adding, subtracting, multiplying, or dividing them requires finding common denominators, factoring, and simplifying by canceling common factors to reduce the expression to simplest form.