How to expand using Pascals Triangle?

#(x+y-1)^2#

1 Answer
George C.
Dec 11, 2017

Pascal's triangle does not help in this case.

Explanation:

Pascal's triangle is useful for expanding powers of binomials, not powers of trinomials.

If the terms of a trinomial are in geometric progression (e.g. #1+x+x^2#) then you can use a variant of Pascal's triangle in which each number is the sum of the three numbers above it, rather than two...

#color(white)(000000000)1#

#color(white)(000000)1color(white)(00)1color(white)(00)1#

#color(white)(000)1color(white)(00)2color(white)(00)3color(white)(00)2color(white)(00)1#

#1color(white)(00)3color(white)(00)6color(white)(00)7color(white)(00)6color(white)(00)3color(white)(00)1#

Unfortunately, in the case of #x+y-1# we have a trinomial which is not in geometric progression, so not even this variant is appropriate.

Instead we can revert to using distributivity, like this:

#(x+y-1)^2 = (x+y-1)(x+y-1)#

#color(white)((x+y-1)^2) = x(x+y-1)+y(x+y-1)-1(x+y-1)#

#color(white)((x+y-1)^2) = (x^2+xy-x)+(xy+y^2-y)-(x+y-1)#

#color(white)((x+y-1)^2) = x^2+y^2+2xy-2x-2y+1#

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