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

1 Answer
Sep 28, 2015

Answer:

#(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.
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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#