Question 806049
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Please help me solve this problem:
DIVIDE x^3-6 by x-2
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        I will show you another, nontraditional way to solve the problem, 

        which is  (I am  100%  sure)  absolutely unexpected to you   (and to many others).



<pre>
Start from this innocent transformation

    x^3 - 6 = (x^3 - 8) + 2.


For the parentheses in the right side, use well known decomposition

    a^3 - b^3 = (a-b)*(a^2 + ab + b^2).


Take here a= x,  b= 2.  You will get then

    x^3 - 8 = (x-2)*(x^2 + 2x + 4).


Therefore,

    x^3 - 6 = (x^3 - 8) + 2 = (x-2)*(x^2 + 2x + 4) + 2.


It means that 

    {{{(x^3-6)/(x-2)}}} = x^2 + 2x + 4  with the remainder of 2.


<U>ANSWER</U>.   {{{(x^3-6)/(x-2)}}} = x^2 + 2x + 4  with the remainder of 2.
</pre>

Solved.



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It is good solution to show it at the local school &nbsp;Math circle.


The students will be positively shocked (= overwhelmed; = stunned) : &nbsp;&nbsp;a deafening effect is guaranteed.


Notice that I did not make any division for it - neither polynomial nor synthetic.


I extracted the solution from the air, &nbsp;using the method called &nbsp;" focus-pocus ".



After such an &nbsp;" exercise ", &nbsp;the students will know and understand Math &nbsp;" in two times better " &nbsp;than before it.


And they will have a stimulus to learn it better.