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Factor the expression completely. (This type of expression arises in calculus in using the “product rule.”)$$ rac{1}{3}(x+6)^{-2 / 3}(2 x-3)^{2}+(x+6)^{1 / 3}(2...

Question

Factor the expression completely. (This type of expression arises in calculus in using the “product rule.”)$$ rac{1}{3}(x+6)^{-2 / 3}(2 x-3)^{2}+(x+6)^{1 / 3}(2)(2 x-3)(2)$$

Factor the expression completely. (This type of expression arises in calculus in using the “product rule.”) $$ \frac{1}{3}(x+6)^{-2 / 3}(2 x-3)^{2}+(x+6)^{1 / 3}(2)(2 x-3)(2) $$



Answers

Factoring Completely Factor the expression completely. This type of expression arises in calculus in using the product rule.
$$3(2 x-1)^{2}(2)(x+3)^{1 / 2}+(2 x-1)^{3}\left(\frac{1}{2}\right)(x+3)^{-1 / 2}$$

We have to completely factor the given expression. 1/3 Time's Expo six to the negative, 2/3 times two Ex My story squared. Plus, that's six to the 1/3 times two times two x minus Three times to you're going to look at both terms and then see what we have in common to figure out our greatest common factor. So in terms of numbers, we got a 1/3 of two and two, so there's no number to factor out there. You got an expert six to the negative 2/3 in the next six to a 1/3 syllabic throughout. This for it. Negative 2/3 term. Finally, we have the two x My history squared under two X minus three to the first power. So just factor out the two x minus three. So we'll start now by factoring goes out and then we'll see what we have left. We'll move from there. So we got an ex post six to the negative 2/3 times two x minus three and then we'll see what we have left. So we still have our 1/3 times two X minus three from our first term from our second term, I have been X plus six times, two times two, and I'll just simplify that two times for no. We can see that for this last part. We can for this and define that first by multiplying it out and then combining the like term soul did not. So Krista V of X plus six to the native. 2/3 times two x minus three and then we have 13 tends to X or 2/3 X minus one plus or X plus 24. Now we just have to combine the like terms. So once again, we start with our X plus six to the negative 2/3 times two X minus three like this, like apprentice ese. And then we have to combine our life term. So 2/3 plus four x becomes 14 over three times X negative. One post tray for is just positive. 23. So our final answer is Expo six to din. End of 2/3 times two x minus three times 14 3rd Exploits. 23

The expression computer. We have no hair to this compression. This is typical. You temple off the factoring crashing over the fractional exponents. Really divided of this two into two parts. First is the ex Crap has three into the power off 21 hours. Three. The second part is the negative. You over three ex graft hands X squared. We're past the through the power of nectar. For all the three they have, they also have the common factor. Thanks, but at past three But the different powers the festival I have the next day. Well, over three and a second. If I had the nectar full over three, we're going to find this modest one. It should be connective for over three because because inactive for three is less than for Is that satanic too? Among all the three. Then we can do the effect factor. Race factor Race factory ization. Ah, we affect her. The specter the factor The X squared past three connective four or three out. Then it becomes the ex grad Uh three. The panel off one manners 203 X square mask. Why it is the, uh The palace becomes the one because the next to you, so always three minors next they'll run all three. It becomes one that objection. The next step is to simplify that. Then these X square past three to Paolo Negative. Four or three They the same and this becomes X squared three minus men is 203 extract. That's that. That would get a final answer. Should be X squared. Plus three. You're the pilot. Power off necked. A full of three terms won over three Ex grad purse. Three. You found two different. If you won't want to simplify for the more you can just checked out the one over three. Then it becomes for this is precious that becomes the X squared plus three. One negative for all three times X squared plus No, it's a final answer. Then we're done.

We have to factor the given expression 1/2 X to the names of 1/2 times three experts for two. The 1/2 plus they has X to the 1/2 times three expose four to the negative one, Huh? So to do this well, first, find a grace common factor, and we'll compare the terms to see what their greatest common factor will be. So you have a 1/2 and, uh, three have so 1/2 the smallest will be using that you have X to the negative one, huh? This is X to the one, huh? So we'll be using X to the native one, huh? We have three expose four to the 1/2 versus the experts. Work to the negative one, huh? So we'll be using the three X plus four to the negative one house. So first rate out the terms that make up our greatest common factor. And then we'll see what gets left behind when we do that to get 1/2 times X didn't give one, huh? Times three x plus four to do negative. 1/2. No one will see what is left. So, for my first dream, you just said that three X plus four there were a second term. We have three times X. We can say we can simply buy in this last prince sees a little bit further by combining like terms. So you get one, huh? Ext is in a grave. One, huh? The explosives for to the native one, huh? Times you can add the three x plus tree ex together, so we get six X plus four. You can actually simplify this parentheses further by factoring out too. And since we'll be multiplying two times, 1/2 that will just read one so well, actually able to cancel out this 1/2 turn. So at Daniel, get excited, and I could have 1/2 times three x plus four to the negative 1/2. And then since we factor out to from that last parentheses will get six divided by two or three X and then four divided by two, which is to. So our final answer is X to the negative one, huh? Times they expose four to the negative 1/2 times three exposed to

All right, so we've been asked to find which of these or expression was we've been given four factor forms on. We need to find which of those is equivalent to this only know me. So since this is a cubic function, it's actually very, very difficult to factor on its own. So at this point, it really would be easier to just test out each of the four expressions we've been given. So we're gonna start with a What we're gonna do is we're just gonna multiply it out and see if it gives us our original polio. So the first thing I'm gonna do is I'm going to just focus on his 1st 2 after aunty's. When I'm lost by that I'm going to do I'm gonna use oil. So foil is first inner Oh, not inner first outer inner and then last. And all that means is that we're gonna multiply the first terms in each parentheses than the most outer terms. The inter terms and then the last terms in each prophecies. So for now, we're just going to ignore the X minus six. So our first terms are the exes. So this is gonna give me X squared than my outer terms. The German most on the left is that X, But your most on the rate is that one. So that's gonna give me ex. My inner terms are the one and the X on the inside, and that's gonna give me that's And in my last terms, in each bottle meal are these two ones, and that is just gonna give me one, and then I'm gonna bring down my X minus six. So now I see that I can combine these two like terms in the middle. So I'm going to do that first. So x square plus two x plus one that I'm going to bring down my X minus suits. So to make this a little bit less confusing, what I like to do when there's three numbers inside of one of my prophecies is that I just focus on this first number and then I'm multiply each of these numbers by that 1st 1 by that X and eight more than negative six Free house. So that's what I'm gonna do. X squared Times X is gonna give me X cute to x Times X is gonna give me a chew X squared and then one times X is gonna give him. That's now I'm going to do the same thing. But this time I'm gonna ignore the X and just look at that negative six. So x squared times negative six is negative. Six excluded. Two x times under six is negative, 12 clicks and then one times negative six is negative. Six. So when I start simplifying, I'm gonna combine like terms. There's only one execute term, so I'm gonna leave that alone. I have a chew X squared and a negative six x squared right from my nose. I get negative or expect, but we can actually stop right here because the question wasn't to foil this out until it's simplified. It was to check and see whether this is equal to our only no meal. We have negative four in front of our X squared term, but it was supposed to be a chew so we can stop right here. We know that a is not correct. Now let's take a look at B. I can actually tell just by looking that B is also not gonna be correct. And the reason that I know that is we just saw with boiling. We first start by multiple ing those two first terms. When I do that, that's gonna give me two x squared times, a whole bunch of things. And then I'll bring down my X minus six. And then after I combined on my leg terms, the next thing I'm gonna do is I'm gonna multiply this chu X squared by the X and that will give me to execute. And these are just the first terms that I'm focusing on. There will be a whole bunch of things after that, but I'm not gonna I'm not actually worried about them because I can see the mistake. Here. I am going to get this as my first term, but there's not supposed to be a two in front of the ex huge, So I know that this one is wrong. So that means I can rule out B next. We're gonna try, see? And we're gonna see if that works. So again, I'm going to start just by focusing on these 1st 2 parentheses, and I'm gonna spoil them out. So first is gonna give me X squared Outer is gonna give me plus X inner is gonna give me three acts and then the last is gonna give me three. And now I'm going to combine these, like, terms that is gonna give me expired, plus or ex plus free times X minus two. And now what I'm going to do is I'm going to do what I did in part a again. I'm just gonna focus on the extra start off with and multiply each of these one by one onto the X. So when I do that, I'm gonna get excuse JJ Plus or at squared Close three X And now I'm gonna ignore the ex. And I'm just gonna look at the negative too. And I'm going to do the same fate. So X squared times and it too, will give me negative two X squared or extends Attitude will give me negative eight X and then three times Negative too. Is negative suits so far? We're looking good. I don't see any problems. So let's keep going. I'll bring down my excuse term because as we've seen time and time again, that first term is always not gonna have any other light terms with it. So we have X cubed. And now, for my expert terms, I have a four expired and a negative chew up square That'll give me two x squared. Then for my extremes, I have positive three X and negative eight x That will give me negative five X And then lastly, I just have a minus six. So there we go. This matches our Kolya meal. So that means the correct answer is C.


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