Divide common factors from the numerator and denominator in...

Divide common factors from the numerator and denominator in the rational expressions in Exercises $19-22 .$ Are the resulting expressions equivalent to the original expressions?
$$
\frac{x^{2}-6 x+5}{x-5}
$$

Key Concepts

Domain Restrictions

Domain restrictions refer to the values that a variable cannot assume in a rational expression, typically those that make the denominator zero. When simplifying by cancelling factors, it is important to note the original restrictions to ensure that the simplified expression has the same domain as the original.

Equivalence of Expressions

Two expressions are considered equivalent if they yield the same value for every input within their shared domain. In the context of simplifying rational expressions, even though the simplified form may look different, it is equivalent to the original expression as long as the appropriate domain restrictions are applied.

Cancellation of Common Factors

Cancellation occurs when a factor that appears in both the numerator and denominator of a rational expression is divided out. This step simplifies the expression, but it must be done carefully since cancellation is only valid for non-zero factors.

Rational Expressions

A rational expression is a quotient of two polynomials where the denominator is not zero. Simplifying these expressions involves ensuring that any cancelled factors are handled correctly to maintain equivalence with the original expression.

Factoring Polynomial Expressions

Factoring is the process of expressing a polynomial as a product of its simpler components. This technique is crucial in simplifying rational expressions because it reveals common factors present in both the numerator and denominator.

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