Simplify each complex fraction. [ 3/(p^2-16) + p ] / [ 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, which can be factored as \( (p-4)(p+4) \).
3. Rewrite the complex fraction: \( \frac{\frac{3}{(p-4)(p+4)} + \frac{p(p-4)(p+4)}{(p-4)(p+4)}}{\frac{1}{p-4}} \).
4. Combine the terms in the numerator over a common denominator: \( \frac{3 + p(p-4)(p+4)}{(p-4)(p+4)} \).
5. Simplify the complex fraction by multiplying by the reciprocal of the denominator: \( \left( \frac{3 + p(p-4)(p+4)}{(p-4)(p+4)} \right) \times (p-4) \).

Key Concepts

Complex Fractions

A complex fraction is a fraction where the numerator, the denominator, or both contain fractions themselves. To simplify complex fractions, one typically finds a common denominator for the inner fractions and rewrites the expression in a simpler form. This process often involves algebraic manipulation to eliminate the nested fractions.

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into simpler components, or factors, that can be multiplied together to yield the original polynomial. In the given question, recognizing that p^2 - 16 can be factored as (p - 4)(p + 4) is crucial for simplifying the complex fraction effectively.

Common Denominators

A common denominator is a shared multiple of the denominators of two or more fractions, allowing for the addition, subtraction, or comparison of those fractions. In simplifying complex fractions, finding a common denominator is essential to combine the fractions in the numerator and simplify the overall expression.