## anonymous 4 years ago The polynomial 1-x+x ^{2}-x ^{3}+...-x ^{15}+x ^{16}-x ^{17} can be written as a polynomial in y=x+1. Find the coefficient of y ^{2}

1. amistre64

i cant make head nor hair of it

2. myininaya

So I guess we should replace x with y-1

3. myininaya

$1-(y-1)+(y-1)^2-(y-1)^3+(y-1)^4 \cdots$ we really don't need to right this any further all of them will be larger degree than y^2

4. myininaya

write*

5. myininaya

that is how i see it anyways

6. Zarkon

you could look at the Taylor polynomial

7. amistre64

doesnt (y-1)^3 expand with a y^2 in it?

8. amistre64

y^3, 3y^2, 3y, 1

9. amistre64

expanding all those terms would create y^2 s

10. myininaya

oh yeah you are right

11. myininaya

that wouldn't be efficient lol

12. myininaya

so we should look at zarkon's way

13. amistre64

well, if its the y^2 coeffs; its the pascal triangle diag then

14. amistre64

2,1,0 ^ 1 3 6 ...

15. amistre64

assuming myins interp is good :)

16. amistre64

..... the L of the pascal .... thats equal to the next ones entry ....

17. amistre64

|dw:1327354588092:dw|

18. Zarkon

you want a polly that looks like this $\sum_{k=0}^{17}a_ky^k$ there $$y=x+1$$

19. Zarkon

*where y=x+1

20. amistre64

1 1* 1 1 2* 1 1 3 3* 1 1 4 6 4* 1 1510 (10) 5 1

21. amistre64

but i think my idea has + and - signs to deal with still

22. Zarkon

I get 816 as the answer

23. amistre64

${{18}\choose {3}}=816.too$

24. asnaseer

I notice that the person you guys are trying to assist has not said a word yet. @kwenisha - does all this make any sense to you? have you attempted it yourself using a different method? any thoughts?

25. Zarkon

using the Taylor polly amounts to computing $\frac{\displaystyle\sum_{k=1}^{17}k(k-1)}{2}$

26. anonymous

@Asnaseer, I am actually trying understand it while working with my partner... I apologize if I come off as just trying to get the answer and not trying to understand it...

27. asnaseer

thats fine @kwenisha - I was just worried in case the experts above left you dazed. :)