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


AoPSWiki:Problem of the Day/June 23, 2011


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

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