Question

How do you use the binomial theorem to expand and simplify the expression `3(x+1)^5-4(x+1)^3`?

Answer 1

The result is `x^5+5x^4+26x^3+18x^2+3x-1`

Details of the expansions follow...

Explanation:

The first binomial will expand into

`ax^5+bx^4+cx^3+dx^2+fx+g`

The trick is coming up with `a, b, c, d, f and g` in as painless a way as possible. You can use Pascal's triangle, or combinations `""_nC_r`, where `n` is 5, and `r` is 0, 1, 2, 3, 4, 5# in that order.

Either way, the coefficients turn out to be 1, 5, 10, 10, 5, 1 in that order.

`x^5+5x^4+10x^3+10dx^2+5x+1`

Now multiply the whole thing by 3.

`3x^5+15x^4+30x^3+30x^2+15x+3`

Subtract from this the second expanded binomial (also get the coefficients in one of the ways mentioned above):

`4(x^3+3x^2+3x+1)`

`4x^3+12x^2+12x+4`

The finished result is found by combining like terms (those in #x^3, x^2, x, etc.)

`x^5+5x^4+26x^3+18x^2+3x-1`

Hope that all makes sense!