- #1

#### socialcoma

(2X+Y)^5

can someone tell me how to expand this?

can someone tell me how to expand this?

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- Thread starter socialcoma
- Start date

- #1

(2X+Y)^5

can someone tell me how to expand this?

can someone tell me how to expand this?

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- #2

Staff Emeritus

Science Advisor

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(2x+y)(2x+y)(2x+y)(2x+y)(2x+y)

Multiply the first two:

(4x

Then the next two, etc. It's messy, but straightforward.

- #3

- #4

- 398

- 0

the Binomial Expansion, also known as Newtons Expansion.

- #5

how do you do Newtons expansion?

- #6

- 322

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Do you know Pascal Triangle? Or have you learned combinations, C^{n}_{r}, before?

- #7

- 937

- 0

Newton's binomial (a.k.a. Newton's expansion) is this:

(a+b)^n=(a^n)+(n*((a^(n-1))*b))+((n*(n-1)*(a^(n-2))*b^2)/(2!))+((n*(n-1)*(n-2)*(a^(n-3))*b^3)/(3!))+...+(b^n)

n can be any rational number

(a+b)^n=(a^n)+(n*((a^(n-1))*b))+((n*(n-1)*(a^(n-2))*b^2)/(2!))+((n*(n-1)*(n-2)*(a^(n-3))*b^3)/(3!))+...+(b^n)

n can be any rational number

Last edited:

- #8

Or shorter

(a+b)^n=SUM (from m=0 to n) C(m out of n)*a^m*b^(n-m)

Damn can I turn on the HTML code?

(a+b)^n=SUM (from m=0 to n) C(m out of n)*a^m*b^(n-m)

Damn can I turn on the HTML code?

Last edited by a moderator:

- #9

This is how it goes from a simple binomial theorem:

(a+b)

(a+b)

(a+b)

(a+b)

As we go on and on we can clearly see that a pattern is emerging. Look at the next post just following this.

- #10

For example in (a+b)

Go to the next coming post.

- #11

For example in (a+b)

As it must become obvious from the above, a starts with power 5 and goes to power 0 and b starts with power 0 and goes to power 5. Of course in the above we are missing the coefficient for each term. Now I show you how to find them.

- #12

i.e. how many of each kind of term:

(a + b)

1 1 0+

0 1 1

-------

1 2 1 0

(a + b)

1 2 1 0+

0 1 2 1

----------

(a + b)

1 3 3 1 0 +

0 1 3 3 1

-------------

(a + b)

1 4 6 4 1 +

0 1 4 6 4 1

---------------------------

a + b)

1 5 10 10 5 1

This is what is known as Pascal's Triangle. The last thing that you have to do is substitute 2x for a and y for b in the above. Good luck

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