In the following exercises, (a) find the LCD for the given...

In the following exercises, (a) find the LCD for the given rational expressions (b) rewrite them as equivalent rational expressions with the lowest common denominator.
$$
\frac{6}{a^{2}+14 a+45}, \frac{5 a}{a^{2}-81}
$$

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Key Concepts

Special Factoring Patterns

Special factoring patterns, such as factoring quadratic trinomials and recognizing the difference of squares, are essential tools in algebra. These patterns allow for the quick and efficient factoring of polynomials, which is a key step in determining the LCD and rewriting rational expressions in a common form.

Rational Expressions

Rational expressions are fractions in which the numerator and/or the denominator are polynomials. Working with these expressions often requires simplifying, finding common denominators, or performing operations like addition and subtraction, which necessitates rewriting them in equivalent forms.

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of its simpler factors, which can be linear or quadratic. This technique is crucial for simplifying rational expressions and finding common denominators when the denominators themselves are polynomial expressions.

Least Common Denominator (LCD)

The least common denominator is the smallest polynomial expression that is a multiple of each of the denominators of given rational expressions. Identifying the LCD typically involves factoring each denominator and combining the unique factors at their highest powers observed across the expressions.

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