Question

6 how does the method change when the example shown involves adding the rational expressions? explain your reasoning.

Explanation:

When adding rational expressions, the core goal is to get a common denominator, but the method adjusts based on the denominators:

1. If denominators are monomials/linear with no factors: Find the least common denominator (LCD) directly, rewrite each expression with the LCD, then add numerators.
2. If denominators are factorable polynomials (like the partially shown $(x+2)(x-2)$): First factor all denominators completely. Identify the LCD as the product of the highest power of each distinct factor. Rewrite each rational expression to have this LCD, then add the numerators and simplify the result (if possible, cancel common factors in the final expression, ensuring to note excluded values that make original denominators zero).

The key change is the requirement to factor denominators first when they are non-linear/factorable, which is not needed for simple, unfactorable denominators. This factoring step is critical to correctly identifying the LCD for more complex rational expressions.

Answer:

When adding rational expressions, the method changes based on the form of the denominators:

• For simple, unfactorable denominators (e.g., monomials, distinct linear terms), you directly find the least common denominator (LCD), rewrite each expression with the LCD, then add the numerators.
• For factorable polynomial denominators (like the partially shown $(x+2)(x-2)$), you first fully factor all denominators (the key added step). You then set the LCD as the product of the highest power of each unique factor. Rewrite each expression using this LCD, add the numerators, and simplify the result (while noting values that make original denominators zero).

The critical change is the mandatory factoring step for complex denominators, which is not required for simple ones, as it is necessary to correctly determine the LCD to add the expressions properly.