Question

How do you use the binomial formula to find expand `(2x+3)^3`?

Answer 1

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

Explanation:

The binomial formula that you need here is

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

Let me colour the same formula to add clarity:

`(color(red)(a) + color(blue)(b))^3 = color(red)(a^3) + 3color(red)(a^2)color(blue)(b) + 3color(red)(a)color(blue)(b^2) + color(blue)(b^3)`

In your case, you would like to expand `(color(red)(2x) + color(blue)(3))^3`, so your

`a = 2x` and `b = 3`

Now, you need to plug `2x` for every occurence of `a` in the formula and plug `3` for every occurence of `b` in the formula:

`(color(red)(2x)+color(blue)(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)`

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

`= 2^3x^3 + 9 * 2^2x^2 + 3 * 2x * 9 + 27`

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