in-13-24-divide-and-express-each-quotient-in-simplest-form-in-each-case-list-any-values-of-the-var-7

Question

In $13-24,$ divide and express each quotient in simplest form. In each case, list any values of the variables for which the fractions are not defined.
$$
\frac{c^{2}-6 c+9}{5 c-15} \div \frac{c-3}{5}
$$

Derrick Hanson · Accepted Answer

Now, before we actually start solving this problem, I see that I&#x27;ve got a quadratic try. No meal in the C squared, minus 60 plus nine. So I want a real quick Just do real quick refresher on how to factor those just in case anybody needed it. Um, that is a quarterback. Try no meal. Meaning it is written in the form a X squared plus B x plus C. Meaning you should look at it as the three terms that it is. You have, ah, squared term. You have the variable to the first power, and then you have a constant that has no variable with it at all. Okay, there are many ways to teach this. The way I teach is I call it the ex meth that there&#x27;s a bunch different ways to show It doesn&#x27;t really matter. I have it as an ex. The idea is, what you&#x27;re doing is you want to take your a times your C, and then you want to find two numbers that multiply to the same value as a time See, but add to your be value you want to side numbers that will multiply to a time. See but we&#x27;ll add to be okay. Meaning, for example, if you look up at this problem here, we have an A value of one, an invisible one. In front of that C squared. We have a B value of negative six that negative six in front of the sea, and then we have a see value in nine. So if I was gonna factor C squared minus six C plus nine, then I&#x27;m looking for two numbers that multiply to nine because one times nine is nine and add to negative 682 numbers that multiply the positive nine. But add to negative six. In this case, that would be negative. Three and negative three right negative times a negative is a positive and negative. Three times negative three would give me a positive nine negative three plus negative three would give me negative six in this case because our A value in this case, because our A value is one. We already have our factor. We can just go straight to C minus three and C minus three. If are a value was not one, we&#x27;d have extra steps, but we don&#x27;t have to deal with that in this problems case. All right, There&#x27;s a quick review on Factoring in general. Um, hopefully that helped. If not, you may to go back and review that stuff because you have learned factoring before this section already. Now, as far as solving this actual problem, Okay, as far as solving this actual problems will get rid of all of this stuff. We do have division going on here. We know we aren&#x27;t supposed to leave it as division. We need to keep our first fraction the same. So we&#x27;ll just rewrite it as c squared. Minus 60 plus nine over five C minus 15. We need to change that division to multiplication and flip or find the reciprocal of our second fraction. Meaning now is five over C minus three. Now we can work on factoring each piece separately and see what cancels out. We already talked about what C squared minus six C plus nine factors to that would be C minus three and C minus three, which, yes, you could also write that as C minus three squared if you wanted to. Okay, five C minus 15 week in G C f. Factor that because five C and 15. Both have five in common. So five c, divided by five would be see minus 15 divided by five would be three. And then we&#x27;ve got times five over C minus three. Everything is factored. So if anything matches up between the numerator and the denominator, we can cancel it out. For example, we have a C minus three in my numerator and a C minus three. In my denominator, those can cancel. We have a C minus three in my numerator and a C minus three and my denominator again. So those canceled. There is a five in the numerator, and there is a five in the denominator, So it was also cancelled, so it looks like literally everything cancelled here. Meeting Our answer is simply one. We do, however, need to still determine any values that could cause this to be undefined. Remember, meaning that if we had zero in the denominator zero on the bottom, that would be an issue. So I have to look at all of my denominators both pre and post reciprocal to make sure I account for all the terms. So here, I&#x27;ve got five C minus 15. Here I have C minus three. So five C minus 15 is the denominator, and we do not want it to equal zero. Hence the slash through the equal sign C minus three is a denominator. And again, we do not want a T equals zero. Well, we saw what happened when we had five C minus 15. We actually factored that five out. So actually, we only have to solve this out once. But we can do it both just to make sure we are confusing anybody. So add 15 over. You&#x27;ve got five. C does not equal 15 divide by five and you&#x27;ve got that see does not equal three. Which is the same thing you would get on the second equation. You would also get See does not equal three meaning. Our final answer is this equation simplifies to one as long as C does not equal positive three

Derrick Hanson · Answer

this is a division problem. So first thing we want to do here is change it into a multiplication problem. Remember, the way we do that first fraction stays exactly the same. We don&#x27;t do anything to change that. We keep that exactly the same. Then we&#x27;re going to change this to multiplication. And the way we&#x27;re going to do that is by flipping or finding the reciprocal of the second fraction. Meaning we will change it to 10 c over three. Be reciprocal. Meaning we just flip it. Okay, now that we have changed it from division to multiplication now we can actually go about trying to factor each of my enumerators and denominators and then cross out anything in common. So six can be factored 22 times three and the bees just there. Five cannot be broken down at all. So we have a five Tennessee 10 can be factor down to two times five. We&#x27;ve got to see three cannot be broken down at all beat. Now that I fully factored each part, I&#x27;m gonna start looking to cancel things out. Anything in the numerator that matches anything in the denominator can cancel out me in between top and bottom. For example. I see that I&#x27;ve got it a three in my numerator and I have a three in my denominator so those can cancel. I noticed that I have a five in my numerator and a five in my denominator so those can cancel. I also see that I have a bee in my numerator and a B in my denominator those can cancel. And finally I have a C and my numerator and to see in my denominator remember, you can Onley cancel between numerator and denominator top and bottom meaning? Yes, I do have two twos, but they are both in their numerator, so they&#x27;re both left behind two times two is four. It looks like I cancelled out all of my denominators, so that would leave me with just a one four divided by one means that my true answer is really just four. Now we&#x27;re not done here because I do have to find any values for variables that would cause me to make this fraction undefined. With division problems, there is a little bit of a catch here. We have two different parts we have to look at. We have to look back at the original equation right here. But we also have to look right here because while we didn&#x27;t change the first fraction, remember, we did find the reciprocal. We did flip that 2nd 1 So now I have to consider this See, as a possible problem, we have to consider this see as a possible problem. And when we flipped the second fraction found its reciprocal. Now we have to worry about this. Be so The question is, let&#x27;s look at five. See first, if we have five c, we cannot have that equal zero. So what value of zero would be a problem? Well, divide five to the other side, and we see, pretty simply see, is not allowed to be zero right 10 c. My other denominator would give me the same problem. Divide by 10 and we just find that C is not allowed to be zero. So we&#x27;ve established that four is our answer, but also see cannot be zero or else it causes this whole thing to be undefined. We also have this three B though that is not able. That does not equal zero. Well, if you divide three over, then you get be does not equal zero, so B and C cannot equal zero. But if they don&#x27;t, our answer is four.

Alisa Lu · Answer

So for this question, were given a fraction 12 over A and were asked to divided by six over for a. So before I begin, I&#x27;m going to say that in this problem cannot be zero, as that would make the denominator equal to zero. And you can&#x27;t divide by zero. Okay, So divided by a fraction is the equivalent operation as multiplying by the reciprocal of the fraction that you&#x27;re dividing by. So this problem told they divided by 648 be rewritten as 12 over a times for a over six. And doing this well, before I multiply across, I&#x27;m going to do some simplification, but and cancellation. So this six will cancel to one, and this will become a two. And here the A&#x27;s will cancel out, and we&#x27;ll just be left with four. Okay, given this will have two times for which is eight over one times one, which is one and eight over one, is just eight. So we have a final answer. It

Derrick Hanson · Answer

so for me personally, any time I see that I have a rational expression as part of my problem, any time that I see have a fraction is part of my problem. I prefer to look at everything on the same level. So I like to have everything representatives fraction because something like this b plus three expression that can cause problems. If you&#x27;re not careful, it&#x27;s purposely not being shown us a fraction just so they can see if they can catch you. Remember if some everything can be expressed as a fraction. So if it doesn&#x27;t look like it&#x27;s a fraction, you can just put it over one, all you have to do is put it over one for it to be a fraction for to be irrational expression. Okay, so if we do that now, I think it becomes much clear how to handle our next step, which is to change this from division to multiplication. Remember, the way you do that is you keep your first fraction exactly the same. Change your division to multiplication, and then you flip or find the reciprocal of your second fraction. Now that I&#x27;ve written it over one, it&#x27;s very clear. I think at least that the reciprocal would be won over B plus three. If you don&#x27;t put that one there, though, it can cause you some issues, I think. OK, so I would just recommend doing that to begin with. Either way, once you&#x27;ve taken care of fighting the reciprocal of your second fraction that changes to multiplication. Now it&#x27;s time to look at each of these individual numerator is and denominators and see if we can factor them down before we start simplifying. Reducing looks like, really the only one that&#x27;s going to be able to simplify is four b plus 12 Can factor, I mean, is gonna be four b plus 12. Four B and 12. Both have a four in common, so let&#x27;s pull that four out. Four. Be divided by four would be be plus 12 divided by four. Would be three Be for my denominator is just that a B one for my numerator is just that a one and B plus three is just that as well. So if I&#x27;m looking for things that are the same between my numerator and denominator, there&#x27;s not a whole lot here. I have the quantity of B plus three in my numerator. And I have the quantity of B plus three in my denominator, so those can cancel out. Other than that, I don&#x27;t see anything else that matches up. So I&#x27;ve got four on top my numerator, because four times one is four, right? And then I just have the be on the bottom. So four Overbey Looks like that&#x27;s our final answer. We do need to find any values that could cause this to be undefined. So remember, you get undefined when zero is in the denominator on the bottom. So I&#x27;m looking at my denominators. That would be be right here, and B plus three right here. Okay, These two fractions here, these two fractions here are just repeated. So I have to solve it out twice. Right? So for my first denominator, B is not allowed to be 00 well, you know, that&#x27;s already solved that awesome. So B does not equal zero, but also be plus three is not allowed to equal zero. To solve that out, I would need to subtract three to the other side and I find that be while not being allowed to be zero is also not allowed to be negative. Three. So my answer is four. Overbey as long as B does not equal zero or negative three.

Alisa Lu · Answer

Okay, So for this question, were given a fraction. A square over a divided by three A over four, and were asked toe do this operation will Also specifying values of a which just couldn&#x27;t work. And those values would be a cannot be zero, as that would make the denominator equal to zero. And we can&#x27;t divide by Sarah. Okay, so before it melts Well, first, um, divided. My fraction is the exact same operation that&#x27;s multiplying by the reciprocal of the fraction you want to divide by. So this could be rewritten as a squared over eight times the reciprocal of the V A over four, just four over three A. And so now go ahead and multiply it cross. Before doing that, I&#x27;m going to do some cancellation simplification. So I notice here that this four and this eight will cancel to give one and two. I also notice that this a in this topping hopes to cancel Eso es will cancel out one of these days and be left with one a. Now, I noticed that this a in this a will also cancel. So this entire ages becomes a one and then way have this um, canceling with to say so. This will also become a one. And so in the end will be left with one times one my collar over two times three, which is 16 where a cannot be.

Alisa Lu · Answer

Okay, So for this question, were given fraction a square plus eight a plus 15 over or a and were asked to divided by from one over a plus three divided by history. So dividing by a plus three is the same as multiplying by the reciprocal day plus three. Just one over a plus three. I&#x27;m so come here. I&#x27;m going to say that our A cannot be equal to zero or negative three, as these values would make the denominator equal to zero when you can&#x27;t divide by zero. So let&#x27;s factor out the numerator here into a plus three and a plus five over for a times one over a plus three. So the eighth Mysteries will cancel. And that just leaves us with one times a post five, which gives you a plus five over for eight times one, which is just for a and so that&#x27;s your final answer

Problem

In $13-24,$ divide and express each quotient in s…

Problem 19 Hard Difficulty

In $13-24,$ divide and express each quotient in simplest form. In each case, list any values of the variables for which the fractions are not defined.
$$
\frac{c^{2}-6 c+9}{5 c-15} \div \frac{c-3}{5}
$$

Answer

$1 ; c \neq 3$

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$1 ; c \neq 3$

Algebra 2 and Trigonometry

Grace H.

Heather Z.

Alayna H.

Michael J.

Absolute Value - Example 1

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Ann Xavier GantertAlgebra 2 and Trigonometry

Grace H.

Heather Z.

Alayna H.

Michael J.

Ann Xavier Gantert

Algebra 2 and Trigonometry