Multiplying & dividing rational expressions
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Divide and express as a simplified rational. State the domain. We start off with this expression. We actually have one rational expression divided by another rational expression. And like we've seen multiple times before, these rational expressions aren't defined when their denominators are equal to 0. So p plus 5 cannot be equal to 0, or if we subtract both sides of this-- we can't call it an equation, but we could call it a not-equation-- by negative 5, if we subtract negative 5 from both sides, you get p cannot be equal to-- these cancel out-- negative 5. That's what that tells us. Over here, we could do the same exercise 4p plus 20 also cannot be equal to 0. If it was, this expression would be undefined. Subtract 20 from both sides. 4p cannot be equal to negative 20. Divide both sides by 4. p cannot be equal to negative 5. So in both situations, p being equal to negative 5 would make either of these rational expressions undefined. So the domain here is the set of all reals such-- or p is equal to the set of all reals such that p does not equal negative 5, or essentially all numbers except for negative 5, all real numbers. We've stated the domain, so now let's actually simplify this expression. When you divide by a fraction or a rational expression, it's the same thing as multiplying by the inverse. Let me just rewrite this thing over here. 2p plus 6 over p plus 5 divided by 10 over 4p plus 20 is the same thing as multiplying by the reciprocal here, multiplying by 4p plus 20 over 10. I changed the division into a multiplication and I flipped this guy right here. Now, this is going to be equal to 2p plus 6 times 4p plus 20 in the numerator. I won't skip too many steps. Let me just write that. 2p plus 6 times 4p plus 20 in the numerator and then p plus 5 times 10 in the denominator. Now, in order to see if we can simplify this, we need to completely factor all of the terms in the numerator and the denominator. In the numerator, 2p plus 6, we can factor out a 2, so the 2p plus 6 we can rewrite it as 2 times p plus 3. Then the 4p plus 20, we can rewrite that. We can factor out a 4 as-- so 4 times p plus 5. Then we have our p plus 5 down there in the denominator. We have this p plus 5. We can just write it down in the denominator. Even 10, we can factor that further into its prime components or into its prime factorization. We can factor 10 into 2 times 5. That's the same thing as 10. Let's see what we can simplify. Of course, this whole time, we have to add the caveat that p cannot equal negative 5. We have to add this restriction on the domain in order for it to be the same expression as the one we started off with. Now, what can we cancel out? We have a 2 divided by a 2. Those cancel out. We have a p plus 5 divided by a p plus 5. We know that p plus 5 isn't going to be equal to 0 because of this constraint, so we can cancel those out. What are we left with? In the numerator, we have 4 times p plus 3, and in the denominator, all we have is that green 5, and we're done! We could right this as 4/5 times p plus 3, or just the way we did it right there. But we don't want to forget that we have to add the constraint p cannot be equal to negative 5, so that this thing is mathematically equivalent to this thing right here.