# Binomial Theorem with coefficients

• January 12th 2013, 02:35 AM
Paze
Binomial Theorem with coefficients
I can get the binomial theorem to work just fine when doing (a+b)^n but it becomes a problem as soon as I put coefficients, e.g.: (2a+3b)^n

Why? What is the difference and how do I calculate with coefficients?
• January 12th 2013, 05:45 AM
veileen
Re: Binomial Theorem with coefficients
Well, there's no difference.

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

$(2a+3b)^n = \sum_{k=0}^{n}\binom{n}{k}(2a)^{n-k}(3b)^k=\sum_{k=0}^{n}\binom{n}{k}2^{n-k}a^{n-k}\cdot 3^kb^k$

Um, did this help?
• January 12th 2013, 08:02 AM
Soroban
Re: Binomial Theorem with coefficients
Hello, Paze!

Quote:

I can get the binomial theorem to work just fine when doing $(a+b)^n$
but it becomes a problem as soon as I put coefficients, e.g.: $(2a+3b)^n$

Why? What is the difference and how do I calculate with coefficients?

Consider $n=3\!:\;(x+y)^3 \:=\:x^3 + 3x^2y + 3xy^2 + y^3$ .[1]

Suppose we are given: . $(2a + 3b)^3$

Then we have: . $\begin{Bmatrix}x \:=\: 2a \\ y \:=\: 3b\end{Bmatrix}$

Substitute into [1]:

. . $(2a + 3b)^3 \;=\;(2a)^3 +3(2a)^2(3b) + 3(2a)(3b)^2 + (3b)^3$

n . . . . . . . . $=\; 8a^3 + 3(4a^2)(3b) + 3(2a)(9b^2) + 27b^3$

n . . . . . . . . $=\;8a^3 + 36a^2b + 54ab^2 + 27b^3$
• January 12th 2013, 02:06 PM
Paze
Re: Binomial Theorem with coefficients
Quote:

Originally Posted by veileen
Well, there's no difference.

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

$(2a+3b)^n = \sum_{k=0}^{n}\binom{n}{k}(2a)^{n-k}(3b)^k=\sum_{k=0}^{n}\binom{n}{k}2^{n-k}a^{n-k}\cdot 3^kb^k$

Um, did this help?

Quote:

Hello, Paze!

I can get the binomial theorem to work just fine when doing
but it becomes a problem as soon as I put coefficients, e.g.:

Why? What is the difference and how do I calculate with coefficients?

Consider .[1]

Suppose we are given: .

Then we have: .

Substitute into [1]:

. .

n . . . . . . . .

n . . . . . . . .

It did indeed help. I was being silly and kept writing a^(n-k)*b^k even though I had coefficients.

Thanks a lot!

Where do I find a guide to write math like you guys do? :)
• January 12th 2013, 02:29 PM
Plato
Re: Binomial Theorem with coefficients
Quote:

Originally Posted by Paze
Where do I find a guide to write math like you guys do? :)

You can use LaTeX tags. Go to this tutorial.

$$x = \neg A\cdot \neg B\cdot C + \neg A\cdot B\cdot \neg C + \neg A\cdot B\cdot C + A\cdot \neg B\cdot \neg C$$ gives $x = \neg A\cdot \neg B\cdot C + \neg A\cdot B\cdot \neg C + \neg A\cdot B\cdot C + A\cdot \neg B\cdot \neg C$
Click on the “go advanced” tab. On the toolbar you will see $\boxed{\Sigma}$ clicking on that give the LaTeX wraps, . The code goes between them.

You can also click on Reply with Quote. That will allow you to ‘steal’ any code that you see that you like.