Question

How do you use the binomial theorem to expand `(4x+3y)^5` ?

Answer 1

`(4x+3y)^5 = 1024x^5+3840x^4y+5760x^3y^2+4320x^2y^3+1620xy^4+243y^5`

Explanation:

The binomial theorem tells us that:

`(a+b)^n = sum_(k=0)^n ((n),(k)) a^(n-k) b^k`

where `((n),(k)) = (n!)/((n-k)!k!)`

The binomial coefficients occur as the `n+1`'th row of Pascal's triangle:

In our example, with `n=5`, the appropriate row of Pascal's triangle is the one beginning `1, 5,...`

`1, 5, 10, 10, 5, 1`

Hence we find:

`(a+b)^5 = a^5+5a^4b+10a^3b^2+10a^2b^3+5ab^4+b^5`

We can then substitute `a=4x` and `b=3y` and expand.

It may be slightly less painful to prepare the coefficients as sequences.

Write powers of `4` in sequence from `4^5` down to `4^0`:

`1024, 256, 64, 16, 4, 1`

Write powers of `3` in sequence from `3^0` up to `3^5`:

`1, 3, 9, 27, 81, 243`

Multiply these two sequences to get:

`1024, 768, 576, 432, 324, 243`

Then multiply by the binomial coefficients to get:

`1024, 3840, 5760, 4320, 1620, 243`

Hence we find:

`(4x+3y)^5 = 1024x^5+3840x^4y+5760x^3y^2+4320x^2y^3+1620xy^4+243y^5`