1. ## Binomial expansion

the coefficient of $x^3$ is larger than $x^2$ by four times....

i can do it easily by drawing Pascal Pyramid.....is there an equation method?

I need help..
Thank you
Bizworldusa

2. ## Re: Binomial expansion

Mmm, look again at your post.

3. ## Re: Binomial expansion

Hello, bizworldusa!

Is my interpretation correct?

The coefficient of $\displaystyle x^3$ is larger than $\displaystyle x^2$ by four times.

I can do it easily by drawing Pascal Pyramid. . Really? . . . How?
Is there an equation method?

From the title, I assume that a binomial expansion is involved
. . and we are to determne the particular binomial.

The simplest would be : .$\displaystyle (ax + b)^3 \;=\;a^3x^3 + 3a^2bx^2 + 3ab^2x + b^3$
. . where $\displaystyle a,b \ne 0.$

Then we have: .$\displaystyle a^3 \:=\:4(3a^2b) \quad\Rightarrow\quad a^3 - 12a^2b \:=\:0$

. . . . . . . . . .$\displaystyle a^2(a-12b) \:=\:0 \quad\Rightarrow\quad a \:=\:12b$

Therefore, one solution is: .$\displaystyle (12x + 1)^3 \:=\:\boxed{1728}\,x^3 + \boxed{432}\,x^2 + 36x + 1$

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Of course, there is an infinite number of solutions.

$\displaystyle (6x+1)^4 \:=\:1296x^4 + \boxed{864}\,x^3 + \boxed{216}\,x^2 + 24x + 1$

$\displaystyle (4x+1)^5 \:=\:1024x^5 + 1280x^4 + \boxed{640}\,x^3 + \boxed{160}\,x^2 + 20x + 1$

4. ## Re: Binomial expansion

Yes your interpretation is correct mr Soroban.