Question

How do I find missing values in binomial expansions?

I have the equation `(1+2x)^a=b+6x+12x^c+dx^3`, an I am asked to find the values of `a`, `b`, `c`, and `d`. How do I find these values?

Answer 1

See a solution process below:

Explanation:

`(1 + 2x)^color(red)(a) = color(blue)(b) + 6x +12x^color(green)(c) + color(purple)(d)x^3`

Because the large exponent on the right side of the equation is `3` then `color(red)(a)` will equal `color(red)(3)`:

`(1 + 2x)^color(red)(3) = color(blue)(b) + 6x +12x^color(green)(c) + color(purple)(d)x^3`

The constant will be `1^3 = 1` therefore: `color(blue)(b)` will be `color(blue)(1)`

`(1 + 2x)^color(red)(3) = color(blue)(1) + 6x +12x^color(green)(c) + color(purple)(d)x^3`

Because this is a binomial the exponent `color(green)(c)` will be `color(green)(2)`

`(1 + 2x)^color(red)(3) = color(blue)(1) + 6x +12x^color(green)(2) + color(purple)(d)x^3`

The coefficient of the `x` term on the left side of the equation will be cube: `2^3 = 8` therefore `color(purple)(d)` will be `color(purple)(8)`

`(1 + 2x)^color(red)(3) = color(blue)(1) + 6x +12x^color(green)(2) + color(purple)(8)x^3`