Question

Expand (1+x+x^2)^3

Answer 1

( 1 + x + x ^ 2 ) ^ 3 = [ ( 1 + x ) + x ^ 2 ] ^ 3

Now use formulas:

( a + b ) ^ 2 = a ^ 2 + 2 a b + b ^ 2

and

( a + b ) ^ 3 = a ^ 3 + 3 a ^ 2 * b + 3 a * b ^ 2 + b ^ 3



[ ( 1 + x ) + x ^ 2 ] ^ 3 =

( 1 + x ) ^ 3 + 3 ( 1 + x ) ^ 2 * x ^ 2 + 3 ( 1 + x ) * ( x ^ 2 ) ^ 2 + ( x ^ 2 ) ^ 3 =

( 1 + x ) ^ 3 + 3 ( 1 + x ) ^ 2 * x ^ 2 + 3 ( 1 + x ) * x ^ 4 + x ^ 6 =

( 1 ^ 3 + 3 * 1 ^ 2 * x + 3 * 1 * x ^ 2 + x ^ 3 ) + 3 * ( 1 ^ 2 + 2 * 1 * x + x ^ 2 ) * x ^ 2 + ( 3 + 3 x ) * x ^ 4 + x ^ 6 =

1 + 3 * 1 * x + 3 * x ^ 2 + x ^ 3 + 3 * ( 1 + 2 x + x ^ 2 ) * x ^ 2 + ( 3 + 3 x ) * x ^ 4 + x ^ 6 =

1 + 3 x + 3 x ^ 2 + x ^ 3 + ( 3 * 1 + 3 * 2 x + 3 * x ^ 2 ) * x ^ 2 + ( 3 * x ^ 4 + 3 x * x ^ 4 ) + x ^ 6 =

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

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

1 + 3 x + 3 x ^ 2 + x ^ 3 + 3 x ^ 2 + 6 x ^ 3 + 3 x ^ 4 + 3 x ^ 4 + 3 x ^ 5 + x ^ 6 =

x ^ 6 + 3 x ^ 5 + 3 x ^ 4 + 3 x ^ 4 + 6 x ^ 3 + x ^ 3 + 3 x ^ 2 + 3 x ^ 2 + 3 x + 1 =

x ^ 6 + 3 x ^ 5 + 6 x ^ 4 + 7 x ^ 3 + 6 x ^ 2 + 3 x + 1

Answer 2

(a+b+c)^3 =
a^3 + 3a^2b + 3a^2c + 3ab^2 + 6abc + 3ac^2 + b^3 + 3b^2c + 3bc^2 + c^3

so, (x^2+x+1)^3 expands as
x^6 + 3x^5 + 3x^4 + 3x^4 + 6x^3 + 3x^2 + x^3 + 3x^2 + 3x + 1

which is what was above