in-3-12-multiply-and-express-each-product-in-simplest-form-in-each-case-list-any-values-of-the-var-8

Question

In $3-12,$ multiply and express each product in simplest form. In each case, list any values of the variables for which the fractions are not defined.
$$
\frac{a^{2}-5 a+4}{3 a+6} \cdot \frac{2 a+4}{a^{2}-16}
$$

Alisa Lu · Accepted Answer

Okay, So for this question were given this fraction he squared minus five plus four over three a plastics and were asked to multiply it by this fractured to a plus four over a square, a minus 16. So before I multiply, I&#x27;m going thio factor where all these numerator Zollo, these denominators s O this will become a minus five for times a minus one over three A plus, two times to a close to over a A plus for a minus four. So now I can go ahead and write, but it values a cannot. Well, so in this problem can&#x27;t be a negative to positive four or negative for as those would make haste, denominators, people to zero. Okay, so now, uh, once again, before I multiply across, I&#x27;m going to do some simple flying, some cross cancellation. So I see a minus four over here, a minus four over here. I can go ahead and cancel these out. I also see a close to share a post to here so I can go ahead and cancel those out as well. Um, so now I think I&#x27;m ready to multiply across. This gives me two times a minus one over three times and plus four. That&#x27;s our finally three

Derrick Hanson · Answer

first things first year. I want to remind myself that to a plus four is a quantity and a squared plus two. A is also a quantity so that I don&#x27;t try and split those things up. Because if it&#x27;s addition or subtraction, that is binding those together. Okay, now, after that, I am looking to simplify each of my four different pieces, my two numerator and my two denominators. I want to simplify each them separately as far as I possibly can. One factor each of them separately as far as a possibly can. And then I looked to cancel things out. For example, starting up in the top left my left numerator to a plus four. I can factor that because to a and four both share a common factor, namely to so I can pull that two out to a divided by two would leave me with a plus four divided by two would leave me with two. In my denominator, six can be split up into two and three and then we&#x27;ve also got the A in my numerator. My second numerator, my right numerator three cannot be split into anything, but I can split a squared into a times a right. That&#x27;s literally what a squared means is a times a finally, my last denominator, my right denominator a squared plus two A. If there is a square term so that means there&#x27;s a whole bunch of different types of factoring that I could do In this case, though, you should mean well, in every case, you should always check for a G C F first before doing any other type of factoring. And in this case, G cf is actually all it is a squared into a both heaven a in common. So I&#x27;m simply going to pull that a out a squared divided by a would leave me with just in a to a divided by a will leave me with two Now that I have fully factored all of my different parts I&#x27;m ready to start canceling things. I see that I&#x27;ve got a two in my numerator and a two in my denominator. So those king cancel I see that I have a three in my numerator and a three in my denominator so those can cancel. I see that I have the quantity of a plus two in my numerator and I have the quantity of a plus two in my denominator. So those two cancelled. I see that I have in a and another A and my numerator and I have a and another A. In my denominator, I have two A&#x27;s, both on top and on bottom. So those canceled. Now this is very interesting because it looks like I have crossed out everything I have here. That&#x27;s not a problem, though. Just remember when you&#x27;re multiplying or dividing, which that&#x27;s what we&#x27;re doing here. When things cancel in multiplication division, that does not mean zero right in addition or subtraction, cancelling out a zero to minus two a zero. But if you&#x27;re multiplying or dividing and things cancel, they cancel toe one right two divided by two is 15 divided by five. Anything divided by itself is one. So when everything cancelled here, that&#x27;s great. That means our answer forgot to change color. That means our answer is legitimately just a nice, simple one. We&#x27;re not done yet because we do also have to take care of any values, Um, for variables for a that could possibly make this undefined meaning, remember, causes to have zero in the denominator. Well looking, my first denominator, I just have six A. Well, that&#x27;s pretty straightforward. If a happens to be zero, that would be a problem, because six times zero is zero and the whole thing is messed up because it&#x27;s undefined. Not a lot of math that you have to do for that. Although I will make this look more like an A over here just to be clear for my second denominator. Now, this one is not quite so clear. Okay, so I&#x27;m going to first stuff. Look at my factored version because that would be was best if I have a squared plus to a set not equal to zero. We need to factor that first, right? So again, I can pull in a out, just like I already did earlier in the problem. That&#x27;s what I&#x27;d actually want to be dealing with. Remember, we have the zero product property that says, if you have two things to quiet is being multiplied to each other, but they both equals zero, or the whole thing is post equal zero. Then that means you can set each piece equal to zero separately or, in our case, not equal to zero. So this one&#x27;s not new. We already knew that a could not equal zero. That&#x27;s a repeat. So not a big deal. Looking at my second piece, though, I would need to subtract two over, and I would get that A does not equal negative, too. And so that needs to be added over here because that is the two. These are the two values we cannot have or also we, because this fraction to be undefined.

Derrick Hanson · Answer

now the first thing I want to do with this problem before start trying to break things down. Cancer or anything like that is I want to put princes around this B plus one, and I would put Prince who&#x27;s around this five b plus five. So I want that to remind me when we&#x27;re multiplying or dividing with rational expressions. Any time I see addition or subtraction, that addition or subtraction is binding. Whatever those two terms are together, I don&#x27;t get to consider them separate things. I don&#x27;t have a B and a one. I have the quantity B plus one. I don&#x27;t have a five being a five. I have the quantity five b plus five, right? We&#x27;re doing multiplying and dividing. So if things are multiplying to each other, we can consider them separate from each other. We are not adding and subtracting right now. If you have things that are being added or subtracted together, those are quantities. They have to stay together. Okay, so I am still going to start off them after that, looking to see if I can ah factor each individual piece right each individual numerator and denominator, and then look to cancel stuff afterwards. Okay, so I have a B plus one. Since that is a quantity, the only thing I can look to mess with that is to see if B and one have anything in common. My greatest common factor that I could pull out. They do not. So I&#x27;m simply just going to rewrite B. Plus One is going to stay the way it is. Four can be broken down into two, and two 12 can be broken down into two to and three at two times two is 44 times three is 12 five B plus five. That is a quantity, but I am able to factor this because five B and five do have one thing in common, namely five. So let&#x27;s use greatest common factor in G C F. To pull that out, I&#x27;ll pull five out of the parentheses, and that would leave me with B plus one because five be divided by five is be and five divided by five is one. So I would have B plus one left inside those parentheses right now that I have fully factored each of my four parts. Now I can look to cancel things that remember, it has to be between the numerator and the denominator. Anything any numerator that matches up with the denominator, I can cancel. All right. For example, I do see that B plus one is here in the numerator, and I also have a B plus one here. The denominator so those can cancel. I see that I have a two here in the denominator, and I have a two here in the numerator. I see that I have a two here in the numerator, and I have a two here in the Dominator, so all of those can cancel with each other. So it looks like now that I don&#x27;t see anything in common between my numerator and denominator, I&#x27;m ready for a final answer. Looks like I have a three left over in my numerator and looks like I have a five left over in my denominator. So my answer would be 3/5. Now, along with that, we&#x27;re also supposed to list any values for variables that would make the fraction be undefined. Remember, we Onley need to look at the denominators, right? You can have zero on top. You can have zero in the numerator. That&#x27;s not a problem. Zero divided by anything is just syrup. But if you have zero in the denominator zero on the bottom, that gives you undefined. So looking at my denominators here, this four is not important to us. It&#x27;s not a variable. There&#x27;s nothing to do with that. So we are entirely focused on this part right here. Five b plus five. Okay, that does have a variable. So the question now is what value of B would cause us to get zero in the denominator. Now the answer is not zero, right? Think about. If I plugged in zero for be five times zero would be zero plus five would be five. So it&#x27;s a different number of that. If you are having trouble, figuring it out will just solve it out. Algebraic Lee. Let&#x27;s take the denominator five b plus five. What is this not allowed to be? Well, we said it&#x27;s not allowed to be zero, so we simply just set that equation equal to zero, and we can solve it out to figure out exactly what we don&#x27;t want be to be right, because we said we don&#x27;t want it to be zero In fact, if we want to be extremely clear, we said we don&#x27;t want it to be zero. So let&#x27;s be even more clear. Let&#x27;s set it not equal to zero right? Were literally showing an outbreak. Representation of what I&#x27;m saying out loud. We don&#x27;t want five B plus five to give a zero, so we&#x27;re just going to say it&#x27;s not supposed to be equal to zero. You still solve this out the same way you solve it at any other time. Subtract five over and you would get five. B is not equal to negative five. Then divide by five and you would get the B should not be equal to negative one that is Thea other part of our answer. If B is negative, one that would cause us to get undefined. You can see if you plug it back into the original equation, so we do not want be to be negative one or really, which they be cannot be negative one and if it&#x27;s not in her answers. 3/5

Alisa Lu · Answer

So for this question, were given a fraction six minus two X over X squared, plus not X squared minus nine times 15 plus five X over for X here. Okay. And then we&#x27;re asked Oh, multiply piece. But before I do that, I&#x27;m just going to go ahead and a factor out, but the numerator and the denominators ticket ups. So I&#x27;m gonna factor out a negative to from this one. Give me X minus three. Uh, and then for the bottom, I&#x27;m going to apply a difference of two squares. Factoring too. Effective this out into X plus stories an X minus three. But this numerator I&#x27;m going to factor out of five to get X plus three over for X. So now, um, I&#x27;m going to go ahead and say that a in this problem cannot be 03 or negative three as thes values will make the denominator equal to zero. And we can&#x27;t have that because you can&#x27;t buy by zero. So now I&#x27;m going to go ahead and do some cancellation and simplification Before I moved quite cross. So I noticed an explosive story in the denominator here explicit during the numerator here Those would cancel out. Um, I also noticed an X minus three over an X minus. Lee, these look also cancel. Well, this negative too. In this four x will. Uh, so if I cancel out the negative tomb into a negative one, this four X will then become to X. And so now we&#x27;re ready to multiply across, and that gives us negative one times five, which is negative. Five over one times two x, which is two X, so our final answer is negative 52

Alisa Lu · Answer

So for this question, were given a square minus 100 over three a times a squared over to a minus 20 and were asked to multiply these two together while also detailing what values a cannot be. In this problem, A cannot be zero or 10 as that would make thes denominators equal to zero. You can have a denominator zero. So, yeah, So before I multiply across, I&#x27;m going to factor this out as well as this denominator. And doing that would give me a minus 10 and eight plus 10. Um, it&#x27;s also important to note that up here this day in one of these days will cancel out. So that means we&#x27;re left with a three and the denominator here times just a a over two times a minus 10. Now, before multiply across, I can do some cross cancellation again. This a mite extended this a minus and we&#x27;ll cancel each other out. And so we left with a times a question over six as our finally

Derrick Hanson · Answer

So we&#x27;re being asked to take 2/3 times 3/4 anytime I&#x27;m multiplying or dividing with rational expressions. What I like to do is I like to start off the problem, looking at it as four individual pieces. I&#x27;ve got numerator here. Numerator here, denominator Here, denominator here. And so if I look at that way, I want to make sure that each individual piece is broken down is factored as far as it can be. Then all look for things that I can start crossing out. Okay, so too to his factor down. That&#x27;s already Prime Three is also primes that covers both the threes. Four, though four can be broken down so that I could rewrite. This is to over three times, 3/2 times two right, cause four can be factored down into its prime factory ization of two times two. And so now that I factored each of the four different pieces down. Now what I&#x27;m gonna look to do is any numerator that matches up with any denominator I can cancel out, and then I&#x27;ll get my final answer would which will be completely reduced. Okay. The idea being you guys know that fraction bars mean the same thing is division. So any numerator here can be cancelled out with any denominator. That&#x27;s the same meaning. I see that I&#x27;ve got a three on top in my numerator and I&#x27;ve got a three on bottom of three denominator, so those you can cancel out. I also see that I have a two in my numerator and I have a two in my denominator so those can cancel out. So I see that all of my new Raiders, everything I had on top canceled out. Remember, when we&#x27;re dealing with multiplication and division, you don&#x27;t cancel out to zero. You cancel out toe one meaning since all my numerator is cancelled out, I have a one for my numerator. In my denominator, I see not everything cancelled out. I do have a single to left over, so I simply bring that across here. And that gives us her answer of 1/2

Problem

In $3-12,$ multiply and express each product in s…

Problem 10 Medium Difficulty

In $3-12,$ multiply and express each product in simplest form. In each case, list any values of the variables for which the fractions are not defined.
$$
\frac{a^{2}-5 a+4}{3 a+6} \cdot \frac{2 a+4}{a^{2}-16}
$$

Answer

$\frac{2(a-1)}{3(a+4)}, a \neq-2,4,-4$

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$\frac{2(a-1)}{3(a+4)}, a \neq-2,4,-4$

Algebra 2 and Trigonometry

Catherine R.

Heather Z.

Caleb E.

Michael J.

Absolute Value - Example 1

Absolute Value - Example 2

Ann Xavier GantertAlgebra 2 and Trigonometry

Catherine R.

Heather Z.

Caleb E.

Michael J.

Ann Xavier Gantert

Algebra 2 and Trigonometry