the result after exploding second degree equation

Question

Why is $$(1+x+x^2)^2 = 1+x^2+(x^2)^2+2(x+x^2+x^3)?$$

My professor wrote that in the class but he didn't explain why.

Thank you

Answer 1

Here is a really long but elementary solution to the problem (does require basic knowledge of factoring by grouping):

$$(a+b)^2 = (a+b)(a+b) = a(a+b) + b(a+b) = a^2 + ab + ab + b^2 = a^2 + 2ab + b^2$$ Now follow the same logic to get $(a+b+c)^2$: $$(a+b+c)^2 $$ $$= (a+b+c)(a+b+c)$$$$ = a(a+b+c) + b(a+b+c) + c(a+b+c)$$ $$ = (a^2 + ab + ac) + (ab+b^2+bc) + (ac+bc+c^2)$$ $$ = a^2+ 2ab + 2ac + b^2 + 2bc + c^2$$ Make the substitutions $a=1$, $b=x$, and $c=x^3$ to get $$(1)^2 + 2(1)(x) + 2(1)(x^2) + (x)^2 + 2(x)(x^2) + (x^2)^2$$ $$= 1 + 2x + 2x^2 + x^2 + 2x^3 + x^4$$ $$= 1 + x^2 + (x^2)^2 + 2(x+x^2+x^3)$$ As desired.

Answer 2

He/she applied the multinomial formula: $$\Bigl(\sum_{i=1}^na_i\Bigr)^2=\sum_{i=1}^na_i^2+2\!\!\!\sum_{1\le i<j\le n}\!\!\!a_i a_j. $$