Question

Binomial Theorem quick help

Answer 1

\[(x+2y)^{10}\]
One term is \[Ax ^{8}y ^{2}\]
How to get A?

Answer 2

kth term in a binomial series: \[\Large (a + b)^{n} is \left(\begin{matrix}n \\ k\end{matrix}\right)a ^{n-k}b ^{k}\]

Answer 3

Here:
a = x, b = 2y, n = 10 and k = 2

Answer 4

So, \[\frac{ 10! }{ 8!2! }\times4x ^{8}y ^{2}= 180x ^{8}y ^{2}\]

Answer 5

Yes!

Answer 6

A = 180

Answer 7

Thanks a lot :)

Answer 8

you are welcome.

Answer 9

Can you help me with a second one?

Answer 10

I will try.

Answer 11

\[\left( \frac{ 2 }{ 3 }x -3\right)^{8}\]
I need to find x to the cube. I just need to understand the start, then I will end by myself

Answer 12

Is the exponent 8 or 3? Can't see it clearly.

Answer 13

To the 8

Answer 14

Compare it to the kth term of the general binomial formula I gave earlier.
a = 2/3x b = -3 n = 8 and what should k be to get the x cube term?

Answer 15

So 5

Answer 16

Ok, lets do this, thanks

Answer 17

Yes. Then you can just plug it into the formula for the kth term and get the x cube term.

Answer 18

Is -4032x3 a realistic result?

Answer 19

I am getting -9072x^3. But haven't double-checked my calculations.

Answer 20

\[\left(\begin{matrix}8 \\ 5\end{matrix}\right)\frac{ 8 }{ 27 }\times(-243)\times x ^{3}\]

Answer 21

and the 8 5 is 56

Answer 22

Yes. I found my mistake. -4032x^3.

Answer 23

Okk. last one, i promise

Answer 24

ok.

Answer 25

I need to find the x4 therm of \[\left( 3x ^{2}-\frac{ 2 }{ x } \right)^{5}\]
So Itis the same thing, right?

Answer 26

A bit tricky because of the x in the denominator.

Answer 27

The exponent of first term will be: 10, 8, 6, 4, 2
The exponent in the denominator of the second term will be: 1, 2, 3, 4, 5
And in the binomial expansion we will be multiplying these two terms with
(10, 8, 6, 4, 2) / (1, 2, 3, 4, 5). Which k value will give you the x^4 exponent?

Answer 28

This is only raised to the fifth power. Why don't you try finding each term in the expansion and write the whole binomial series for this problem?

Answer 29

k will go from 0 to 5. Find each term in the expansion.

Answer 30

Ok, but I dont have time tonight, I will do it tomorrow and write here my results.

Answer 31

Thanks a lot :D really

Answer 32

sure.

Answer 33

If you don't want to write out all the 6 terms we can figure out what value of k will have the x^4 term.

As k goes from 0 to 5, the first term will have the following exponents and it will be in the numerator:
10, 8, 6, 4, 2, 0
The corresponding exponents of the second term will be in denominator:
0, 1, 2, 3, 4, 5
y^m/y^n = y^(m-n). So we just need to subtract the numerator exponents and the denominator exponents to see which one yields x^4:
(10-0), (8-1), (6-2), (4-3), (2-4), (0-5)
It is the third term that will give us x^4. And remember k starts from 0 so the third term will have k = 2

So here we have a = 3x^2 ; b = (-2/x) ; n = 5 and k = 2