Question

How do you use the binomial theorem to expand `(2sqrtt-1)^3`?

Answer 1

`(2sqrtt-1)^3=8tsqrtt-12t+6sqrtt-1`

Explanation:

To find the binomial coefficients, I will use Pascal's triangle:
`color(white)(aaaaaaaaaaaaaaaaaaa)1`
`color(white)(aaaaaaaaaaaaaaaaa)1color(white)(aaa)1`
`color(white)(aaaaaaaaaaaaaaa)1color(white)(aaa)2color(white)(aaa)1`
`color(white)(aaaaaaaaaaaaa)1color(white)(aaa)3color(white)(aaa)3color(white)(aaa)1`

Looking at the 4th row (mind you, we're counting from `0`), we can see that the coefficients will be `1,3,3,1`.

The degree of the left term decreases from left to right and the degree of the right term increases from left to right. This means our expansion will be:
`(2sqrtt-1)^3=(2sqrtt)^3+3(2sqrtt)^2(-1)+3(2sqrtt)(-1)^2+(-1)^3=`

`=8tsqrtt-12t+6sqrtt-1`