Maths Olympiad Prep

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Algebra Difficulty 4.8 AIME Find the answer

Example 3. Factorize: a5+a+1a^{5}+a+1.

Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly.

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solve the equation (a5−a2)+(a2+a+1)=(a2+a+1)(a3−a2+1).\begin{aligned} & \left(a^{5}-a^{2}\right)+\left(a^{2}+a+1\right) \\ & =\left(a^{2}+a+1\right)\left(a^{3}-a^{2}+1\right) .\end{aligned}

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.