Simplify each rational expression. (6 x^2+13 x-63)/(12...

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Key Concepts

Rational Expressions

A rational expression is a quotient of two polynomials. Simplification involves rewriting the expression in reduced form by factoring the numerator and denominator and canceling any common factors while noting any restrictions on the variable that would make the denominator zero.

Polynomial Factoring

Polynomial factoring is the process of expressing a polynomial as a product of lower-degree polynomials. In the context of simplifying rational expressions, factoring quadratics into linear factors is particularly important as it often reveals common factors in the numerator and the denominator.

Common Factor Cancellation

Once the numerator and denominator are factored, any common factors can be canceled to simplify the expression. However, it is crucial to ensure that the values for which these factors are zero are excluded from the domain of the expression.

Domain Restrictions

When simplifying rational expressions, it is important to identify and state the restrictions on the variable. These restrictions arise from values that would cause the original denominator to be zero, since division by zero is undefined.

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