Expand (1 + x +2x^2)^6 upto and including x^3?

Question

Expand (1+x+2x^2)^6 in ascending powers of x upto and including x^3

I have tried to go around this problem by

expanding (1+y)^6

This gives me 1 + 6y + 15y^2 + 20y^3 +15y^4 +

6y^5 + y^6

I then substitute x + 2x^2 into the expansion

as y. I do this upto 20y^3

This gives 1 + 6x + 12x^2 + 15x^2 + 60x^4 +

20x^3 + 160x^6

I then simplify this to give my final answer of

1 + 6x + 27x^2 + 20x^3

I belive this is correct but online binomial

expansion calculators say the answer should be

1 + 6x + 27x^2 + 80x^3

I dont understand why i am not getting 80x^3

Please help?

Answer 1

For the 15y^2 term, be more careful when you square y. That's where your error lies. You should have both a 60x^3 and a 60x^4 term.

But congrats on picking a good approach and almost getting it right the first time!