5

3"5 -4 ~3 ~2 -1 0 1 2 3 4 5...

Question

3"5 -4 ~3 ~2 -1 0 1 2 3 4 5

3 "5 -4 ~3 ~2 -1 0 1 2 3 4 5



Answers

$$\left\{(3,-1),(5,0),(0,5),\left(4, \frac{2}{3}\right)\right\}$$ $$\{(1,-3),(2,-7),(4,-3),(5,-5)\}$$

Here. I'm gonna square both numbers. So 3/5 developed by nine Over 100. So have something. 3/5 times. 100 over nine. So I can divide Thies to buy three and get one in three to buy these to buy If I didn't get one and 20 So I have 20/3.

So in this problem we're given this five x 5 matrix And were asked to use a matrix calculator, luckily in order to determine or find the determinant. All right, because you can do this by hand. I might be a little difficult. You can probably pick Grow four here and you know avoid two of the calculations. But then you're gonna still be left with four x 4 Matrix sees and working those ways where's goes down. So this can be a lot quicker. So we go over here to the new matrix. In our matrix calculator. We went to Desmond's dot com math tools matrix calculator to get this. And our matrix is a five by five. I gotta get this thing built up here, the framework of it five by five. and so the first entry is a three and then a minus two A four three. You know, one, Then a 1-0 -1 zero 2102 on Euro. And then 5 -15. My ass. One 032 zero three, two Than 4 7 -8. 4 seven. Mine it's eight and zero zero. And then we have 123 one two three 02 zero two chris Sanders. So that's all in there. Now I need the determinant of this Five by five. So I go d et determinant of A And there it is 410. That was a lot quicker and a lot easier. And trying to do this all out my hand, wasn't it

If we want to evaluate the equation above, we can use the binomial theorem written ingredient. When the equation is in this form, we can see from each individual term that X in the binomial theorem corresponds toe one for a in the binomial theorem corresponds 23 over four. An end in the binomial theorem corresponds to five. We know that the binomial theorem is used to expand equations of the form X plus a the end. So we know that this entire equation here can be rewritten as 1/4 plus 3/4 to the fifth. This equals one to the fifth, which just equals one.

All right. So for the following question, we have a very long equation that we need to find a numerical value outs. I'm gonna radio now. It's may take a little bit. It's it is very long. So obviously as we can see, this is already a binomial expansion, and hopefully we all know the formula for that. But if lot luckily, this question so long that, um, you can probably find it by telling done writing this whole thing out, you know, we're gonna persevere. Write it all out. It's just easier when you write it out. It's a better way of learning. Should always right your questions out whenever doing it. The sword is a doozy. Still going a couple more lines and we're gonna be good. Some of you can probably already kind of guess the rest of them or not. Guess story. No, the rest of them a pattern or formula wherever we like to call it should be a four. Uh, and this is the last line. So all this we need to figure out what it equals. So we know that from the binomial theorem, whenever we're given an X plus a why to be X to the Y, and it could be equal to the sun of and J is equal starts at zero of an over J and then we have X to the n minus j. And then why Teoh the J So when we actually expand that out in just purely, um, mathematical form, it looks like this. So and over zero x to the end, plus and over one x to the n minus one. Why plus and over to extra the war and Linus to why? To the power of to and then plus, And it goes on essentially until you reach end over end. So in this case, we're gonna look for our aunts, RJ's and our, um, our xer wise and our ends essentially. So from the first term, you can see that it's gonna be end over zero and X to the end. So in this case, I was gonna point some things out. I think that five is our and it's our end, Um, and then from the first formula, and then this right here, the 1/4 is gonna be our X. And then, um, for the next one, we have that we once again have our end of five and then one is the next step in the formula. We once again have this as X still, and and now we have this which is actually gonna be our Why. So now I'm just gonna write out everything about we have. We have X is equal to 1/4. Why is equal to 3/4 and we have that end is equal to five. So the formula that would have started this binomial theorem would look like this 1/4, plus 3/4 to the power of five, and that's actually going to be a formula. So now, since there's no variables in it, we could just solve for the number in this case. So 1/4 plus 3/4 is actually is going to be one. It's gonna be one to the power five, and obviously that equals one. So therefore the equation slash binomial expansion. I'm just going right. B E for short is equal to one. And we found that by looking at the formula that were given and kind of reverse engineering to get it back to the formula that we know so we could solve for a numerical value


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