91Ó°ÊÓ

Factoring polynomials is a key step in simplifying rational expressions. It involves rewriting a polynomial as a product of its factors. Think of factors as building blocks that multiply together to form the original polynomial.
For example, let's look at the polynomial \( x^2 - 4 \). This can be rewritten using the difference of squares formula as \((x + 2)(x - 2)\).
When you factor the denominator, \( x^2 - x - 6 \), it becomes \((x - 3)(x + 2)\).

To effectively factor polynomials:

• Identify common formulas like difference of squares or trinomials.
• Look for patterns or common terms you can group.
• Use trial and error for finding numbers that multiply and add up correctly in trinomials.

By breaking down expressions into their factors, we make it easier to simplify the overall expression in the next steps.

Once the polynomials are factored, the next step is to identify any common factors shared between the numerator and the denominator. Common factors allow us to simplify the expression because multiplying by a factor is analogous to dividing by that same factor.
In our example, after factoring, the numerator is \((x + 2)(x - 2)\) and the denominator is \((x - 3)(x + 2)\).
Here, \((x + 2)\) is common to both the numerator and the denominator.

Simplification process:

• Cancel the common factors from both the top and the bottom.
• Ensure that the expression remains unchanged in value, only simplified in form.

The result of canceling the common factor in this case gives us the simplified form \( \frac{x - 2}{x - 3} \).
This process is a shortcut to making expressions simpler without altering their value.

Excluded values are an essential concept to keep in mind after simplifying a rational expression. These are values that would make the original denominator equal to zero, which is undefined in mathematics because division by zero is not possible.
In the expression \((x - 3)(x + 2)\), the denominator would become zero if the values of \( x \) are substituted to be \( 3 \) or \( -2 \).

Steps to determine excluded values:

• Set each factor in the denominator equal to zero.
• Solve these equations for the variable.

For this example, setting \( x - 3 = 0 \) gives \( x = 3 \) and \( x + 2 = 0 \) gives \( x = -2 \).
These values need to be highlighted because they are not part of the domain of the expression—meaning \( x \) can be any number except these. It's important to state them even after simplification, to maintain the expression's integrity.