Simplify the rational expression. (x^2-x-12)/(x^2+5 x+6) |...

Name: Problem 20, Chapter 0: Algebra and Trigonometry
Uploaded: 2019-06-26T23:05:58-08:00
Duration: 3 min 37 s
Channel: Taylor Shimono
Description: Simplify the rational expression. (x^2-x-12)/(x^2+5 x+6)

Key Concepts

Polynomial Factoring

This concept involves expressing a polynomial as a product of its simpler binomial or monomial factors. It is particularly useful for simplifying rational expressions because it allows one to identify and cancel common factors from the numerator and the denominator, provided that they do not lead to division by zero.

Simplifying Rational Expressions

Simplifying rational expressions is a process where a complex fraction is reduced to its simplest form by factoring the numerator and the denominator and canceling any common factors. This process is analogous to reducing numerical fractions by dividing both the numerator and denominator by their greatest common factor.

Domain Restrictions

When simplifying rational expressions, it is essential to consider the values of the variable that would make any denominator zero. These restrictions are important because they indicate the values for which the original expression is undefined, even if cancellation seems to remove the problematic factor.

Transcript

00:01 So for this problem, we are going to be simplifying this expression right here.

00:05 And even though nothing jumps out at us as being really easy to cancel right off the bat, we can tell that both our numerator and our denominator are going to be factorable.

00:16 So i'll go ahead and start by factoring the numerator using my diamond and rectangle method, and i'll go ahead and do that in red.

00:24 So we start with the top of our diamond is going to be the coefficient of our x squared term, 1 and our loan term that is not a coefficient of x which is going to be negative 12 so we'll multiply those together negative 12 times 1 will give us negative 12 and the bottom of our rectangle will be given by the coefficient of the single x term which is going to be negative 1 so now we know to fill out the rest of this diamond we are going to have to find two numbers that multiply to be negative 12 and add to be negative 1.

01:03 And those two numbers happen to be negative 4 and positive 3.

01:08 So our next step is going to be making a rectangle with our original term.

01:17 That's the x squared term from our original expression.

01:22 And then we'll have our two terms from our diamond, negative 4x and positive 3x in these next two boxes.

01:31 And the bottom right box is going to be the final term in our original expression, which is negative 12.

01:38 And then we are going to go ahead and go down each column and across each row and find the greatest common factor in each of those.

01:51 So in our first column, we get x.

01:54 Second column, we have negative 4.

01:57 And the first row we again have x and the second row we will have a positive three so now that we've factored our numerator we can go ahead and do the same thing with our denominator which i'll do in blue so for the top of our diamond we're going to multiply six times one which is six and the bottom we will have our five and the terms that are going to add to be five and multiply to be six are going to be positive 3 and positive 2...

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