Simplify each rational expression. Also, list all numbers that mu...

Question

Simplify each rational expression. Also, list all numbers that must be excluded from the domain. [x^3+2x^2]/[x+2]

Accepted Answer

Hello today we're going to be simplifying a rational expression and finding the restrictions of its domain. So the rational expression given to us is four X to the fourth Power plus eight X cubed all of that over X squared minus four. Now in order to find the restrictions of the domain, we need to solve for X and figure out what values cause dysfunction to become undefined. In order to do that we're gonna take our denominator which is X squared minus four and set it not equal to zero. Now what we want to do is go ahead and solve for X. In order to solve for X. We start by adding four to both sides. That will leave us with X squared equal to four. And now we want to go ahead and take the square root of both sides. Now keep in mind that every time you take a square root it gives you two values positive and negative and the square root of four is just gonna give us to so X cannot equal positive two and X cannot equal negative two. So that's gonna be our restrictions. So that's the first part of our solution. Now what we want to do is go ahead and simplify this rational expression, let's take a look at the numerator. The numerator is four X to the 4th power plus eight x cubed. Now one thing to note about this both of these terms have a multiple of four so we can go ahead and factor out of four and both of these terms have at least X cubed inside of it. So we can go ahead and factor out X cubed. So if we factor out four X cubed, what we're left with is four X cubed times X plus two. Now let's go ahead and take a look at our denominator. Our denominator is x squared minus four. Now this is a fact arable quantity because this is a difference of square terms. Recall that the difference of squares a squared minus B squared factors to a plus B times a minus B in this case here are A is going to be X and r. B is going to be four. So our denominator is going to factor two, X plus two times x minus two. Now if we combine all of that together, what we're left with is four X. Skewed times X plus two over X plus two times X minus two. And if you notice we have X plus two in both the numerator and denominator so we can go ahead and cancel out those quantities. What we're left with is four X cubed Over X -2. And that's gonna be the simplest form of this rational expression. So that would be our solution. So our solution is four X cubed over X minus two with the restrictions of the of the domain being positive two and negative two. That means that the solution to this problem is going to be C. So I hope this video helps you work with rational expressions and I'll go ahead and see you all in the next video.

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