Question

Simplify each expression. $$ \frac{x-2}{x^{2}-4} $$

   Simplify each expression.
$$
\frac{x-2}{x^{2}-4}
$$

Algebra A Combined Function

Algebra A Combined Function

Elayn Martin-Gay 4th Edition

Chapter 7, Problem 29

Instant Answer

Solved by Expert Victor Salazar

02/07/2022

Step 1

The expression $x^{2}-4$ is a difference of squares, which can be factored into $(x-2)(x+2)$. So, the expression becomes: $$ \frac{x-2}{(x-2)(x+2)} $$ Show more…

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Simplify each expression. $$ \frac{x-2}{x^{2}-4} $$

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Key Concepts

Domain Restrictions

When simplifying rational expressions, it is important to account for the restrictions on the variable. These restrictions ensure that the canceled factors do not correspond to values that would have made the denominator zero in the original expression, thus preserving the expression's validity.

Cancellation of Common Factors

Once the polynomials in the numerator and denominator are factored, any common factors can be canceled, provided the values of the variable do not make the original denominator zero. This step reduces the expression to its simplest form.

Rational Expressions

A rational expression is a fraction in which both the numerator and the denominator are polynomials. Simplifying rational expressions typically involves reducing the fraction by canceling common factors and ensuring that the expression is in its simplest form.

Factoring Polynomials

Factoring is the process of expressing a polynomial as a product of its factors. This is a crucial step in simplifying rational expressions as it unveils common factors in the numerator and denominator that can be canceled.

Difference of Squares

The difference of squares is a specific factoring pattern given by a^2 - b^2 = (a - b)(a + b). Recognizing this pattern is essential when dealing with quadratic polynomials that fit this form, as it simplifies the expression significantly.

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