Question 1210271
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If x(x - 3) = - 1, then the value of x³(x³ - 18) will be
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<pre>
We are given that value of x satisfies equation

    x*(x-3) = -1.           (1)


In other words, x is the root of the polynomial  

    p(x) = x^2 - 3x + 1.    (2)


They ask to find the value of polynomial  Q(x) = x^3*(x^3-18) = x^6 - 18x^3.


To do it, make long division and find the REMAINDER of division Q(x) by p(x)

    Q(x) = {{{(x^4 + 3x^3 + 8x^2 + 3x + 1)*(x^2-3x+1)}}} - 1.    (3)


Thus, the remainder is -1.


    It means, that if you substitute the root x of the polynomial  p(x) = x^2 -3x + 1 

    into polynomial Q(x), then the first addend in the right side of (3) will be 0 (zero),

    and you will get the final value of -1 for Q(x).


<U>ANSWER</U>.  The value of  {{{x^3(x^3 - 18)}}}  is -1.
</pre>

Solved.



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<H3>The post-solution note</H3>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Polynomial &nbsp;p(x) &nbsp;is a quadratic, &nbsp;so it has two roots.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Therefore, &nbsp;you would expect that polynomial &nbsp;Q(x) &nbsp;has two different possible values - but we got only one value, &nbsp;-1.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Why ?


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;It is because the remainder of division &nbsp;Q(x) &nbsp;by &nbsp;p(x) &nbsp;is not a linear polynomial, &nbsp;as we could expect,

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;but a single constant value -1, &nbsp;instead.


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Therefore, &nbsp;it produces a unique single value &nbsp;-1 &nbsp;for &nbsp;Q(x).