Question 1209688
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I made a plot, using the plotting tool DESMOS at www.desmos.com/calculator,
which is free of charge online plotting tool for common use.


See the plot under this link

https://www.desmos.com/calculator/07tfoacyi9


The plot shows absence of integer roots.


<H3>The CONCLUSION and the DIAGNOSIS</H3>This post is not a Math problem, but a FATAL GIBBERISH, instead.



/\/\/\/\/\/\/\/\/\/\ &nbsp;&nbsp;Second attempt &nbsp;&nbsp;/\/\/\/\/\/\/\/\/\/\


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The polynomial

f(x) = x^3 + 10x^2 + 21x + 10 + 4x^3 - 17x^2 + 8x - 66

has one integer root.  What is it?
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<pre>
1. Combine like terms:
   f(x) = (x^3 + 4x^3) + (10x^2 - 17x^2) + (21x + 8x) + (10 - 66)
   f(x) = 5x^3 - 7x^2 + 29x - 56

2. Rational Root Theorem: This theorem states that any rational root of the polynomial must be of the form p/q, 
   where p is a factor of the constant term (-56) and q is a factor of the leading coefficient (5).

   Possible values for p: ±1, ±2, ±4, ±7, ±8, ±14, ±28, ±56
   Possible values for q: ±1, ±5

   Possible rational roots: ±1, ±2, ±4, ±7, ±8, ±14, ±28, ±56, ±1/5, ±2/5, ±4/5, ±7/5, ±8/5, ±14/5, ±28/5, ±56/5

   Possible integer roots: ±1, ±2, ±4, ±7, ±8, ±14, ±28, ±56.


3. I checked all these possible integer roots using my MS Excel.

   NO ONE of this values is a root of the polynomial 5x^3 - 7x^2 + 29x - 56.
</pre>

<H3>It means that your problem is a FAKE: it DECEIVES a reader.</H3>

<U>ANSWER</U>.  &nbsp;&nbsp;This problem is a &nbsp;FAKE : &nbsp;&nbsp;it deceives a reader and presents a &nbsp;FALSE &nbsp;situation as if it is true.

         The given polynomial &nbsp;HAS &nbsp;NO &nbsp;integer roots.  &nbsp;&nbsp;No one integer number is a root of this polynomial.


<H3>The answer in the post by @CPhill is INCORRECT (= WRONG).</H3>