SOLUTION: If {{{ x^3+5x-10=0 }}} Then find the value of {{{ x^7+100x^2+25x }}}

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Question 1178846: If
Then find the value of

Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52832)   (Show Source): You can put this solution on YOUR website!
.
If   x^3 + 5x - 10 = 0,   then find the value of   x^7 + 100x^2 + 25x.
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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.


ANSWER.  If  x^3 + 5x - 10 = 0,  then  x^7 + 100x^2 + 25x = 250.

Solved.

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

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

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


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


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Do not forget to post your  "THANKS"  to me for my teaching.



Answer by greenestamps(13203)   (Show Source): You can put this solution on YOUR website!


Given: x^3+5x-10=0

Rewrite as x^3=10-5x, and also as x^3+5x=10

Square both sides in the first of those: x^6=100-100x+25x^2

Multiply by x: x^7=100x-100x^2+25x^3

Use that to evaluate the expression we are to evaluate:

x^7+100x^2+25x = 100x+25x^3+25x = 125x+25x^3 = 25(x^3+5x) = 25(10) = 250

ANSWER: 250


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