Question 1178846
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If    x^3 + 5x - 10 = 0,    then find the value of    x^7 + 100x^2 + 25x.
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<pre>
From  x^3 + 5x - 10 = 0  we express

      x^3 = 10-5x.     (1)


Next, we consider x^7  and will transform it step by step, decreasing the degree of x,

replacing  x^3 at each appearance by  (10-5x), according to (1)


    x*7 = x^4 * x^3 = x^4 * (10-5x) = 10x^4 - 5x^5 = 10x*x^3 - 5x^2*x^3 = 

        = 10x*(10-5x) - 5x^2*(10-5x) = 100x - 50x^2 - 50x^2 + 25x^3 = 

        = 100x - 100x^2 + 25*(10-5x) = 100x - 100x^2 + 250 - 125x = -100x^2 - 25x + 250.    (2)


Now  x^7 + 100x^2 + 25x = substitute expression (2) instead of x^7 = 

     = (-100x^2 - 25x + 250) + 100x^2 + 25x = combine like terms = 250.


<U>ANSWER</U>.  If  x^3 + 5x - 10 = 0,  then  x^7 + 100x^2 + 25x = 250.
</pre>

Solved.


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It looks like a trick, &nbsp;a focus, &nbsp;but, actually, &nbsp;it is &nbsp;THE &nbsp;METHOD.


In mathematical language, &nbsp;it is called &nbsp;&nbsp;"decreasing a degree", &nbsp;or &nbsp;"lowering a degree".


We systematically use expression  &nbsp;(1),  &nbsp;&nbsp;x^3 = 10-5x,  &nbsp;&nbsp;to decrease the degree of  &nbsp;&nbsp;x^7, &nbsp;step by step.



Having this expression &nbsp;(1), &nbsp;it allows us to run/(to start)/(to launch) &nbsp;the &nbsp;"decreasing a degree" &nbsp;engine.



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