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

Jun 1, 2016

${\left(1 + x + {x}^{2}\right)}^{3} = 1 + 3 x + 6 {x}^{2} + 7 {x}^{3} + 6 {x}^{4} + 3 {x}^{5} + {x}^{6}$

#### Explanation:

I don't think you do use the binomial theorem for this one, since $\left(1 + x + {x}^{2}\right)$ is a trinomial, not a binomial.

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

$1 \textcolor{w h i t e}{00} 1 \textcolor{w h i t e}{00} 1$
$\textcolor{w h i t e}{000} 1 \textcolor{w h i t e}{00} 1 \textcolor{w h i t e}{00} 1$
$\underline{\textcolor{w h i t e}{000000} 1 \textcolor{w h i t e}{00} 1 \textcolor{w h i t e}{00} 1}$
$1 \textcolor{w h i t e}{00} 2 \textcolor{w h i t e}{00} 3 \textcolor{w h i t e}{00} 2 \textcolor{w h i t e}{00} 1$

$1 \textcolor{w h i t e}{00} 2 \textcolor{w h i t e}{00} 3 \textcolor{w h i t e}{00} 2 \textcolor{w h i t e}{00} 1$
$\textcolor{w h i t e}{000} 1 \textcolor{w h i t e}{00} 2 \textcolor{w h i t e}{00} 3 \textcolor{w h i t e}{00} 2 \textcolor{w h i t e}{00} 1$
$\underline{\textcolor{w h i t e}{000000} 1 \textcolor{w h i t e}{00} 2 \textcolor{w h i t e}{00} 3 \textcolor{w h i t e}{00} 2 \textcolor{w h i t e}{00} 1}$
$1 \textcolor{w h i t e}{00} 3 \textcolor{w h i t e}{00} 6 \textcolor{w h i t e}{00} 7 \textcolor{w h i t e}{00} 6 \textcolor{w h i t e}{00} 3 \textcolor{w h i t e}{00} 1$

So:

${\left(1 + x + {x}^{2}\right)}^{3} = 1 + 3 x + 6 {x}^{2} + 7 {x}^{3} + 6 {x}^{4} + 3 {x}^{5} + {x}^{6}$

Note that this is like picking the $n$th row of a variant of Pascal's triangle in which each number is the sum of three numbers above it:

$\textcolor{w h i t e}{0000000000000} 1$
$\textcolor{w h i t e}{0000000000} 1 \textcolor{w h i t e}{00} 1 \textcolor{w h i t e}{00} 1$
$\textcolor{w h i t e}{0000000} 1 \textcolor{w h i t e}{00} 2 \textcolor{w h i t e}{00} 3 \textcolor{w h i t e}{00} 2 \textcolor{w h i t e}{00} 1$
$\textcolor{w h i t e}{0000} 1 \textcolor{w h i t e}{00} 3 \textcolor{w h i t e}{00} 6 \textcolor{w h i t e}{00} 7 \textcolor{w h i t e}{00} 6 \textcolor{w h i t e}{00} 3 \textcolor{w h i t e}{00} 1$