AoPS Wiki talk:Problem of the Day/June 23, 2011

Problem

Since the first term is $x^5$ , the two polynomials that it factors into must either be $x^2$ and $x^3$ , or $x$ and $x^4$ .

Due to the constant term being $1$ , the two factors must both have $1$ or $-1$ as their constants.

Starting off with $x^2+1$ and $x^3+1$ , with product $x^5+x^3+x^2+1$ , we need a way to get rid of the $x^3$ and $x^2$ terms in the product.

Adding a $-x^2$ into $x^3+1$ , so it becomes $x^3-x^2+1$ , gives a product of $(x^3-x^2+1)(x^2+1)=x^5-x^4+x^3+1$ .

To get rid of the $-x^4$ and the $x^3$ term, we change $x^2+1$ to $x^2+x+1$ . The new product is $x^5+x+1$ , which is what we are trying to factor.

Therefore, $\boxed{x^5+x+1=(x^3-x^2+1)(x^2+x+1)}$ .