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Find the integer $x$ that satisfies the equation $x^5 - 101 x^3 - 999 x^{ 2 } + 100900 = 0$.

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Answer:

$10$

Step by Step Explanation:

1. We need to find the integer value of $x$ that satisfies the equation $x^5 - 101 x^3 - 999 x^{ 2 } + 100900 = 0.$
2. $\begin{align} & x^5 - 101 x^3 - 999 x^{ 2 } + 100900 = 0 \\ \implies & x^5 - 101 x^3 - 999 x^{ 2 } + 100899 + 1 = 0 \\ \implies & x^3(x^{ 2 } - 101) - 999(x^{ 2 } - 101) + 1 = 0 \\ \implies & (x^{ 2 } - 101)(x^3 - 999) + 1 = 0 && \ldots (1) \\ \end{align}$
3. We observe that the only integer that satisfies $(1)$ is $10.$
   Hence, the value of $x$ is $10.$

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