Simplify each complex fraction. [ y + 1/(y²-9) ] / [ 1/(y + 3) ]

Verified step by step guidance

1. Identify the complex fraction: \(\frac{y + \frac{1}{y^2 - 9}}{\frac{1}{y + 3}}\).
2. Recognize that \(y^2 - 9\) is a difference of squares and factor it as \(y^2 - 9 = (y - 3)(y + 3)\).
3. Rewrite the numerator by combining the terms over a common denominator: \(y + \frac{1}{(y - 3)(y + 3)} = \frac{y(y - 3)(y + 3)}{(y - 3)(y + 3)} + \frac{1}{(y - 3)(y + 3)}\).
4. Combine the fractions in the numerator: \(\frac{y(y - 3)(y + 3) + 1}{(y - 3)(y + 3)}\).
5. Divide the numerator by the denominator fraction by multiplying by the reciprocal: \(\frac{y(y - 3)(y + 3) + 1}{(y - 3)(y + 3)} \times (y + 3)\), then simplify the expression by canceling common factors.

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 products like the difference of squares, e.g., y^2 - 9 = (y - 3)(y + 3), helps simplify expressions and cancel common factors in fractions.

Division of Fractions

Dividing by a fraction is equivalent to multiplying by its reciprocal. To simplify a complex fraction, rewrite the division as multiplication by flipping the denominator fraction, then multiply numerators and denominators accordingly.