Question

How do I use the the binomial theorem to expand `(t + w)^4`?

Answer 1

`(t+w)^4=t^4+4t^3w^1+6t^2w^2+4t^1w^3+w^4`

Explanation:

First, you should get to know Pascal's triangle. Here's a small portion of it up to 6 lines. Basically you're just adding the numbers that are beside each other, then writing the sum below them. (For example, `1+2` in the third line equals to `3` in the fourth line.) If you want a better explanation for that, feel free to do some more research on it.

Okay, so take a look at the triangle. The numbers in each line are the coefficients of the terms when a binomial is raised to a certain power. The first line is for `n^0`, the second is for `n^1`, the third is for `n^2`, etc.

For `(t+w)^4`, we'll use the 5th line (1 4 6 4 1). Let's write it down:
`1tw+4tw+6tw+4tw+1tw`.

Now, for the first term, the exponent of the first variable `t` will be `4` and it will descend in the next terms. The exponent of `w` will start at `0` and ascend in the next terms.
`1t^4w^0+4t^3w^1+6t^2w^2+4t^1w^3+1t^0w^4`

Simplify:
`t^4+4t^3w^1+6t^2w^2+4t^1w^3+w^4`