Simplify the rational expression. (x^2-x-2)/(x^2-1) |...

Name: Problem 18, Chapter 0: Algebra and Trigonometry
Uploaded: 2019-06-26T23:01:29-08:00
Duration: 3 min 19 s
Channel: Taylor Shimono
Description: Simplify the rational expression. (x^2-x-2)/(x^2-1)

Key Concepts

Domain Restrictions

Domain restrictions refer to the values of the variable that would make the denominator zero. When simplifying rational expressions, it is crucial to note these restrictions even if the problematic factors are canceled during the simplification process.

Cancellation of Common Factors

Cancellation of common factors is the process of removing identical factors from the numerator and the denominator. This step simplifies the expression but requires caution as it introduces restrictions where the canceled factor must not be equal to zero.

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. They are simplified by factoring the polynomials and canceling any common factors, as long as those factors are not zero, to reduce the expression to its simplest form.

Polynomial Factoring

Polynomial factoring involves rewriting a polynomial as a product of simpler polynomials or factors. In the process of simplifying rational expressions, factoring the numerator and the denominator is essential to identify and eliminate common factors.

Transcript

00:01 So for this problem, we are asked to simplify our expression given to us as x squared minus x minus 2 over x squared minus 1.

00:11 So we can see right away that nothing is going to cancel out immediately, but we know that both our numerator and our denominator are going to be factorable using our diamond and rectangle method.

00:25 So i'll go ahead and start by factoring the numerator, and i'll do that in red.

00:30 So we know that in our diamond, the top is going to be the coefficient of the first term and the last term multiplied together.

00:37 So that'll be our 1 in front of our x squared, and our negative 2, which multiplies to be negative 2.

00:44 And the bottom of our diamond will be this negative 1 coefficient right there.

00:49 And now we want to find two numbers that are going to add to be negative 1 and multiply to be negative.

00:55 2.

00:56 So we know that we will have negative 2 and positive 1.

01:01 And we can go ahead and make our rectangle next.

01:05 And the top left corner of our rectangle is going to be our x squared from the original equation.

01:13 Then we'll have minus 2x and plus 1x.

01:17 That's from our diamond.

01:19 And finally we will have our minus 2 and that comes again from our original expression.

01:25 So we are going to be our going to go ahead and pull out our factors, our greatest common factor from each of the rows and columns...

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