Dividing rational expressions

divide and express as a simplified rational state the domain so 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 the denominators are equal to zero so P plus 5 cannot be equal to 0 if we subtract both sides of this of this I guess we can't call it equation we could call it the not equation by negative 5 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 and over here we could do the same exercise for P plus 20 also cannot be equal to 0 if it was this expression would be undefined subtract 20 from both sides for P 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 expressions either of these rational expressions undefined so the domain here the domain is the domain is all we could say the set of all reals such or PP 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 so we've stated the domain now let's actually simplify this expression so when you divide by a fraction or a rational expression that's the same thing as multiplying by the inverse so let me just rewrite this thing over here so 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 change 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 and although 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 five times ten 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 in the denominator and in the numerator 2p plus six we can factor out a two so the 2p plus six 2p plus six we can rewrite it as two times P plus three and then the four P plus twenty we can rewrite that we can factor out a four as so times four four times P plus five and then we have our P plus five down there in the denominator we have this P plus five we can just write it down in the denominator P plus five and then even ten we can factor that further until we can into its prime components or in its prime factorization we can factor ten and factor ten in two times two times five that's the same thing as ten now let's see what we can simplify and 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 P plus 5 we know that people's 5 isn't going to be equal to 0 because of this constraint so we can cancel those out so what are we left with in the numerator we have 4 times P plus 3 times P plus 3 and in the denominator all we have is that green 5 all we have is that 5 and we're done we could write this as 4/5 times P plus 3 or just the way we did it right there but we don't want to forget 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