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(e) Using Binomial expansion, show that (1 +9' #+I= 128 1024 (5 marks) (b) By substituting a suitable value of x into your answer in part (a), show that V5 114...

Question

(e) Using Binomial expansion, show that (1 +9' #+I= 128 1024 (5 marks) (b) By substituting a suitable value of x into your answer in part (a), show that V5 1145 ZIS (5 marks)3_

(e) Using Binomial expansion, show that (1 +9' #+I= 128 1024 (5 marks) (b) By substituting a suitable value of x into your answer in part (a), show that V5 1145 ZIS (5 marks) 3_



Answers

Find the term of the binomial expansion containing the given power of $x$. $$(x+1)^{8} ; x^{5}$$

Okay, so it's separating the following as a plus and they could be to the power of five. And let's expend we have. And which is five? She's a girl agent are fine. March is one. A four on a to B. What, you still that's going to be a three? You could have beat it apart. You Then we have four of Tuesday A to a negative three on a beach in power today. Five. Choose for A and B to the power of four. And then lastly, we have five juice five B to the power related 3 to 4 or five. So it's simplified us. We get a five minus five, a four b and then we have five shoes. Two. That's a 10. A three beat you and then we have minus five juice. Three. That's a ton. A squared. You killed, aren't you? Score is a five year plus five. A before and an minus, maybe to the power of five

We wish to use the final meal serum to ride out this expansion. So here is a Pascal's triangle, which we will use to find the binomial coefficients easily so we can see here our power is five. So that is our end, Which means the rule that we're interested in is this well, by over here. So let's use that and write down their coefficient. So that will be one five tongue tang. Bye. And what next? We're going to fill in the rest. So the first one that's going to be the first terms of power event. So our first term here is 38 38 to the power on which is but next one is the first term. So that's 38 and that's the power of four times in the second term. Be careful because we have a minus. So which means it's actually plus negative for B, which means our second term is actually negative. Four b. This is going to be negative for B to the power of one with this 10 over just a little bit. So we have more room. So then plus, and the next term is going to be the first term, just three a will be to the power of three and then make it a four b to the power of to and then create or power to and negative four b to the power one. Nope. Sorry, not 13 This is supposed to increase in power. The next one is going to be three A. So just will decrease in power that the power of one and make it affordable. Increase in power. That's the power for And the last term is going to be 38 apart zero, which is just one. So we don't need to write it down. And then negative four b will be the power of size And now we just have to simplify this. So we have three to the power of five that's going to be 243 a to the power five and then we have negative four times three to the power of four times five, which is negative 16 to 0 and we have eight of the four needed. The one the next one is going to be three cubed times Negative. Four squared times 10. That's our co efficient. So that is 4320 and we have a to the power 3 ft of the power to next one is going to be negative. Four to the power of three times three to the power of two times this China in the front. So which means our coefficient is negative. But 760 this will be a squared and be cute. Next time we have negative four to the power of four and in times three times five. So that is 3840. We have a to the power of one and B to the power of four. And the last one is negative. Four to the power of life, which is negative. 1024. We don't have any age, but we do have be power effect. So there you have it. This is our binomial expansion.

Even a plus B square power trial. And by former is we could use some mention under try juice K A about Truman sk and people Juba que case down from zero trail. They want to find a time continuing down be our eight Indians. Then we want to have this. Now will be on a farm. People, Are you okay? We want to have it UK in country great and four k meson. Could you far? The volatile we're looking for is a form Trager's for a power off the trail. Honest, Full, be power eight And then we get Nico Jew drowned, which is far. You got your 495? A About it be about eight.

So we want to write the 1st 3 terms of the binomial a minus to be to the 15th Power. We're gonna use our binomial theorem two determine these 1st 3 terms. And so, if you look, um, we can use our formula where eight of the end is gonna be the first term in over one eyes the coefficient for our second term and then accent minus one. Why is our second term and then here is our third term. So just gonna need to plug in our values in replace a or ex with our first term. X is also gonna be this a term why will be the to be so our 1st 1 we're gonna have a and is 15. So there's our first term. That one's fairly easy. Then we're gonna add 15 one coefficient and then X again is a and minus. One would be 14. And our why value is gonna be negative to be, and then we're gonna add 15 over tube coefficient, uh, again, X is a and that's gonna be to the 13th power and then negative to be squared. So our first term still is eight of the 15 plus And then our combination 15 1 we can use her calculator. I'm just gonna delete here. Put 15 comma one and again, that's 15. But we have Thio Multiply it by Negative too. So that actually becomes negative. 30. So let's Ah, eraser minus sign there too, as well. It's again. This was 15 and then we have negative too. So becomes negative. 38 of the 14th be plus and then we have the combination. 15 2 I'm gonna just go back to my calculator Here, plug in the two. When we see it's 100 and five, that's 105 and negative two squared becomes four. So 105 times four does become 420. So we're gonna add 420 eight of the 13 b squared And again, here's our first term second term third term


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