How do we expand #(y+2)^4# using Pascal's triangle?

1 Answer
Nov 8, 2017

#(y+2)^4=y^4+8y^3+24y^2+32x+32#

Explanation:

An expansion of #(x+y)^n# contains #(n+1)# terms syarting from #x^n#, then #x^(n-1)y#, #x^(n-2)y^2#, #x^(n-3)y^3# and so on and finally #y^n#. Observe that while power of #x# reduces from #n# to finally #0# that of #y# increases from #0# to #n# and sum of the powers of the two always remains #n#.

Further their coefficients follow the numbers in #n^(th)# row of Pascal's triangle (shown below).

To build Pascal's triangle, we start with "1" at the top, then continue placing numbers below it in a triangular pattern with #1# at both the ends. Inside each number is the sum of the numbers directly above it.

http://ptri1.tripod.com/

Hence #(x+y)^4=x^4+4x^3y+6x^2y^2+4xy^3+y^4#

and hence #(y+2)^4# is

#y^4+4y^3*2+6y^2*2^2+4x*2^3+2^4#

or #y^4+8y^3+24y^2+32x+32#