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Factoring quadratic expressions is an essential skill in algebra. A quadratic expression is generally of the form \(ax^2 + bx + c\). The goal is to rewrite it as the product of two binomials. For instance, consider the expression \(z^2 + 18z + 81\). We look for two numbers that multiply to 81 (the constant term) and add up to 18 (the coefficient of the middle term). These numbers are 9 and 9, so we factor the expression to get \((z + 9)(z + 9)\). Similarly, for \(z^2 + 7z - 18\), we find factors of -18 that add up to 7, which are 9 and -2. Thus, \(z^2 + 7z - 18\) factors to \((z + 9)(z - 2)\). This factorization step makes it easier to manipulate and simplify rational expressions. Always double-check your factors by expanding the binomials to ensure correctness.

Dividing rational expressions involves several steps, the key one being to multiply by the reciprocal of the divisor. For the expression \(\frac{z^2 + 18z + 81}{z^2 + 7z - 18} \div \frac{z^2 - 5z + 6}{z^2 - 4z + 4}\), once we have factored all quadratic expressions, our job is to rewrite the division as a multiplication. This is achieved by flipping the second fraction and changing the division sign to multiplication. Specifically, we convert: \[\frac{(z + 9)(z + 9)}{(z + 9)(z - 2)} \div \frac{(z - 2)(z - 3)}{(z - 2)(z - 2)} \rightarrow \frac{(z + 9)(z + 9)}{(z + 9)(z - 2)} \times \frac{(z - 2)(z - 2)}{(z - 2)(z - 3)}\]. This important step simplifies dealing with the algebraic expressions moving forward.

In simplifying rational expressions, you often encounter terms that can cancel each other out. Common factors are factors that appear in both the numerator and denominator. After rewriting our expression as a multiplication problem, \(\frac{(z + 9)(z + 9)}{(z + 9)(z - 2)} \times \frac{(z - 2)(z - 2)}{(z - 2)(z - 3)}\), we see common factors: \(z + 9\) and \(z - 2\). We systematically cancel these out: \[\frac{(z + 9)\cancel{(z + 9)}\cancel{(z - 2)}(z - 2)}{\cancel{(z + 9)}\cancel{(z - 2)}(z - 2)(z - 3)}\]. What remains is \(\frac{(z + 9)(z - 2)}{(z - 2)(z - 3)}\). Notice that \(z - 2\) still appears in both numerator and denominator, and cancelling one more time, we get \(\frac{z + 9}{z - 3}\). Canceling common factors reduces the complexity of the expression, making it more straightforward to work with.

When multiplying rational expressions, we multiply the numerators together and the denominators together. After converting a division into a multiplication problem and canceling common factors, we might end up with a simpler expression. Before canceling in our example, we had: \(\frac{(z + 9)(z + 9)}{(z + 9)(z - 2)} \times \frac{(z - 2)(z - 2)}{(z - 2)(z - 3)}\). Multiplying these directly, we'd combine the numerators and denominators: \[\frac{(z + 9)(z + 9)(z - 2)(z - 2)}{(z + 9)(z - 2)(z - 2)(z - 3)}\]. Simplification by canceling common factors is essentially the same as simplifying fraction multiplication. The end result, through these simplifications, yields \(\frac{z + 9}{z - 3}\). Always simplify your result as much as possible to get the final answer.