Question

BINOMIAL THEOREM question: http://prntscr.com/8ib6g8
Find the numerically greatest coefficient in the expansion of (2 - x^2/4)^10.

Answer 1

I'm confused on what to do with the negative sign??
I know we use the formula:
\[\frac{ T _{k+1} }{ T_{k}} = \frac{ n - k + 1 }{ k } \times \frac{ b }{ a }\]

Answer 2

Or do we ignore the negative? O_O

Answer 3

2^10

Answer 4

I don't know what this is, so I don't know what they mean. I think the binomial is

\[(a+b)^n = \sum_{k=0}^n \binom{n}{k}a^kb^{n-k}\]

So I'm not so sure what they're wanting here, I am guessing coefficient on x after it's all expanded out right?

Answer 5

This is what i did but i'm not sure if i'm solving this correctly :S
http://prntscr.com/8ib6g8

Answer 6

it would be -1280

Answer 7

if negatives dont matter then that is right i think

Answer 8

I know this doesn't really help with the problem all that much,
but I would factor the 1/4 out of each term,
that is just getting in the way in my opinion,\[\large\rm \left(2-\frac{x^2}{4}\right)^{10}=\frac{1}{4^{10}}\left(8-x^2\right)^{10}\]

Answer 9

Yah I would imagine that's what they meant by "numerically greatest", like ignore the sign :O that sounds right.

Answer 10

sheesh why cant they just say magnitude or abs value -.-

Answer 11

hmm i'm lost xD
I don't know whether it will be -1280 or 1280...
I think maybe it'll be -1280??

Answer 12

why x=1? :o
does that just make the rest of the calculations easier or something?

Answer 13

i actually have no idea LOLL.
I just got taught to set x = 1 if there was no given value of x?

So unless the question gave a specific value of x, you'd just plug in x = 1 :/
If they asked for the term independent of x, then you set x = 0

Answer 14

I don't know that weird T formula 0_o
I'mma do it the long way and see if that checks out.\[\large\rm \frac{T_{k+1}}{T_k}=\frac{\color{orangered}{\left(\begin{matrix}10 \\ k+1\end{matrix}\right)}2^{10-(k+1)}\left(-\frac{x}{4}\right)^{k+1}}{\color{orangered}{\left(\begin{matrix}10 \\ k\end{matrix}\right)}2^{10-k}\left(-\frac{x}{4}\right)^{k}}\]Which I guess simplifies down a bit right? Ummm\[\large\rm \frac{T_{k+1}}{T_k}=\frac{\color{orangered}{\frac{10!}{(k+1)!(10-(k+1))!}}\left(-\frac{x}{4}\right)^1}{\color{orangered}{\frac{10!}{k!(10-k)!}}2^1}\]And more simplifying -_-\[\large\rm \frac{T_{k+1}}{T_k}=\frac{k!(10-k)!}{(k+1)!(10-(k+1))!}\cdot\left(-\frac{x}{8}\right)\]and further -_-\[\large\rm \frac{T_{k+1}}{T_k}=\frac{(10-k)}{(k+1)}\cdot\left(-\frac{x}{8}\right)\]Hmmmm yah I'm seeing how you got 11 :O
Thinking...

Answer 15

1280 i read some post where numerical greatest means the magnitude of the number

Answer 16

Not seeing how you got 11* blah

Answer 17

pls show me appreciation with your medals

Answer 18

is there even a difference between "greatest coefficient" and "numerically greatest coefficient" ? o.o
But alrighties, i guess i'll stick with 1280...

Answer 19

Oh your formula works actually,\[\large\rm \frac{T_{k+1}}{T_k}=\frac{(10-\color{royalblue}{(k)})}{(\color{royalblue}{k}+1)}\cdot\left(-\frac{x}{8}\right)\]
\[\large\rm \frac{T_{k+1}}{T_k}=\frac{(10-\color{royalblue}{(k-1)})}{(\color{royalblue}{k-1}+1)}\cdot\left(-\frac{x}{8}\right)\]
\[\large\rm \frac{T_{k+1}}{T_k}=\frac{10-k+1}{k}\cdot\left(-\frac{x}{8}\right)\]Ok I'll simmer down XD

Answer 20

yes greatest coefficient means the biggest number where positive numbers are greater than the negative numbers

Numerically greatest means the number that is farthest away from 0 or the origin

Answer 21

LMAO Zepdrix xD Your dedication is so strong. Binomials give me headaches.

Dan, wait, so if it is the number farthest away from 0 or the origin, wouldn't it be -1280?

Answer 22

yes thats right

Answer 23

okaaayyy ^_^
Thank you ALL~ <3.

Answer 24

no problem

Answer 25

noob

Answer 26

-.-

Answer 27

ooo snap

Answer 28

.

Answer 29

*