Question

use pascals triangle to expand the binomical expression (1+2x)^3?

Answer 1

Let's write first few lines of Pascal's triangle:

$~~~~~~~~~~~~1\\ ~~~~~~~~1~~~~~~~1\\ ~~~~1~~~~~~2~~~~~~~1\\ 1~~~~~~3~~~~~~~3~~~~~~1$

We can see that coefficients for cubic binomial are 1, 3, 3, 1.

$(1+2x)^3=1\cdot1^3+3\cdot1^2\cdot2x+\\ +3\cdot1\cdot(2x)^2+1\cdot(2x)^3=\\ =1+6x+12x^2+8x^3$