Question

Simplify each rational expression. $$\frac{x^{2}-2 x+1}{2-x-x^{2}}$$

   Simplify each rational expression.
$$\frac{x^{2}-2 x+1}{2-x-x^{2}}$$

Precalculus : Building Concepts and Connections

Precalculus : Building Concepts and Connections

Revathi Narasimhan 1st Edition

Chapter 3, Problem 3

Instant Answer

Solved by Expert Ramzi Deek

05/03/2020

Step 1

The expression $x^{2}-2x+1$ can be written as $(x-1)^{2}$. So, the expression becomes: $$\frac{(x-1)^{2}}{2-x-x^{2}}$$ Show more…

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Simplify each rational expression. $$\frac{x^{2}-2 x+1}{2-x-x^{2}}$$

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Key Concepts

Rational Expressions

A rational expression is a quotient of two polynomials. Simplifying such expressions involves rewriting the numerator and denominator in factored form so that any common factors can be cancelled, streamlining the expression while keeping in mind restrictions on the variable.

Polynomial Factoring

Factoring is the process of breaking down a polynomial into a product of simpler polynomials (or factors). This technique is essential in simplifying rational expressions, as it reveals common factors in the numerator and denominator that can be reduced.

Cancellation of Common Factors

Once the numerator and denominator are factored, common factors may appear in both parts of the rational expression. Cancelling these factors simplifies the expression, but one must be careful to note that such cancellation is only valid when the common factors are not equal to zero.

Domain Restrictions

When simplifying rational expressions, it is important to consider the values for which the expression is undefined—typically when the denominator is zero. Any cancellation of factors does not remove the original restrictions, so the variable must be constrained to values that do not lead to division by zero.

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