How do you use the binomial formula to expand #[x+(y+1)]^3#?

1 Answer
Jun 15, 2016

Answer:

#x^3+y^3+3x^2y+3xy^2+3x^2+3y^2+6xy+3x+3y+1#

Explanation:

This binomial has the form #(a+b)^3#
We expand the binomial by applying this property:
#(a+b)^3=a^3+3a^2b+3ab^2+b^3#.

Where in given binomial #a=x# and #b=y+1#

We have:
#[x+(y+1)]^3=#
# x^3+3x^2(y+1)+3x(y+1)^2+(y+1)^3# remark it as (1)

In the above expand we still have two binomials to expand
#(y+1)^3# and #(y+1)^2#

For #(y+1)^3# we have to use the above cubed property
So #(y+1)^3=y^3+3y^2+3y+1#. Remark it as (2)

For #(y+1)^2# we have to use the squared of the sum that says:
#(a+b)^2 =a^2+2ab+b^2#

So #(y+1)^2=y^2+2y+1#. Remark it as (3)

Substituting (2) and (3) in equation (1) we have:

#x^3+3x^2(y+1)+3x(y+1)^2+(y+1)^3#
#=x^3+3x^2(y+1)+3x(y^2+2y+1)+(y^3+3y^2+3y+1)#
#=x^3+3x^2y+3x^2+3xy^2+6xy+3x+y^3+3y^2+3y+1#
We have to add the similar terms but in this polynomial we don't have similar terms , we can arrange the terms .
Thus,

#[x+(y+1)]^3=x^3+y^3+3x^2y+3xy^2+3x^2+3y^2+6xy+3x+3y+1#