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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