Question

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

Answer 1

`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`