Factor using the Binomial Theorem? #8a^3+12a^2b+6ab^2+b^3#
1 Answer
Mar 27, 2018
Explanation:
Note that
So let's expand
The binomial theorem tells us that:
#(x+y)^n = sum_(k=0)^n ((n),(k)) x^(n-k) y^k#
where
In particular:
#(x+y)^3 = ((3),(0))x^3+((3),(1))x^2y+((3),(2))xy^2+((3),(3))y^3#
We find:
#((3),(0)) = ((3),(3)) = (3!)/((3!)(0!)) = 1#
#((3),(1)) = ((3),(2)) = (3!)/((2!)(1!)) = 6/2 = 3#
So:
#(x+y)^3 = x^3+3x^2y+3xy^2+y^3#
Then putting
#(2a+b)^3 = (2a)^3+3(2a)^2(b)+2(2a)(b)^2+(b)^3#
#color(white)((2a+b)^3) = 8a^3+12a^2b+6ab^2+b^3#