# How do you expand #(1+x+x^2)^3 # using the binomial theorem?

##### 1 Answer

#### Explanation:

I don't think you do use the binomial theorem for this one, since

We can long multiply the coefficients a couple of times like this:

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

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

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

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

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

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

#underline(color(white)(000000)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#

So:

#(1+x+x^2)^3=1+3x+6x^2+7x^3+6x^4+3x^5+x^6#

Note that this is like picking the

#color(white)(0000000000000)1#

#color(white)(0000000000)1color(white)(00)1color(white)(00)1#

#color(white)(0000000)1color(white)(00)2color(white)(00)3color(white)(00)2color(white)(00)1#

#color(white)(0000)1color(white)(00)3color(white)(00)6color(white)(00)7color(white)(00)6color(white)(00)3color(white)(00)1#