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

1 Answer
Jan 1, 2018

Answer:

#(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#