Question

How do you find the binomial expansion for `(2x+3)^3`?

Answer 1

`(2x+3)^3 = 8x^3 + 36x^2 + 54x + 27`

Explanation:

With the Pascal's triangle, it's easy to find every binomial expansion :

Each term, of this triangle, is the result of the sum of two terms on the top-line. (example in red)

`1`
`1. 1`
`color(blue)(1. 2. 1)`
`1. color(red)3 . color(red)3. 1`
`1. 4. color(red)6. 4. 1`
...

More, each line has the information of one binomial expansion :

The 1st line, for the power `0`
The 2nd, for the power `1`
The 3rd, for the power `2`...

For example : `(a+b)^2` we will use the 3rd line in blue following this expansion :

`(a+b)^2 = color(blue)1*a^2*b^0 + color(blue)2*a^1*b^1 + color(blue)1*a^0*b^2`

Then : `(a+b)^2 = a^2 + 2ab + b^2`

To the power `3` :

`(a+b)^3 = color(green)1*a^3*b^0 + color(green)3*a^2*b^1 + color(green)3*a^1*b^2 + color(green)1*a^0*b^3`

Then `(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3`

So here we have `color(red)(a=2x)` and `color(blue)(b=3)` :

And `(2x+3)^3 = color(red)((2x))^3 + 3*color(red)((2x))^2*color(blue)3 + 3*color(red)((2x))*color(blue)3^2 + color(blue)3^3`

Therefore : `(2x+3)^3 = 8x^3 + 36x^2 + 54x + 27`

Answer 2

`(2x+3)^3=8x^3+36x^2+54x+27`

Explanation:

`(2x+3)^3`

Use the cube of a sum method, in which `(a+b)^3=a^3+3a^2b+3ab^2+b^3`.

`a=2x;` `b=3`

`(2x+3)^3=(2x)^3+(3*2x^2*3)+(3*2x*3^2)+3^3` =

`8x^3+(3*4x^2*3)+(3*2x*9)+27` =

`8x^3+(9*4x^2)+(27*2x)+27` =

`8x^3+36x^2+54x+27`