Question 1180967
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This is more polynomials but I could not find a section for that. 
im sorry if there is. {{{highlight(cross(its))}}} <U>It is</U> my first time on here.

for what real values of k does the equation x^3 - 2x^2 - 4x + k = 0 have at least one root strictly between -2 and 0?
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<pre>
If you rewrite the equation in this EQUIVALENT form,

    x^3 - 2x^2 - 4x = -k,


then you can re-formulate the question in this EQUIVALENT form


    +-------------------------------------------------------------- ------+
    |              for what real values of k the equation                 |
    |    x^3 - 2x^2 - 4x = -k  has at least one root between -2 and 0 ?   |
    +---------------------------------------------------------------------+


Now look into this plot of the polynomial  p(x) = x^3 - 2x^2 - 4x.



    {{{graph( 400, 400, -5, 5, -10, 5,        
              x^3 - 2x^2 - 4x
)}}}


                   Plot y = x^3 - 2x^2 - 4x 



From this Figure, identify that part of the plot, which is over the interval (-2,0).


Over this interval, the plot makes a downward cup.


The values of "-k"  (note: "minus k")  we are seeking for, are the values on y-axis  of the polynomial p(x) over the interval (-2,0).



    So, now it is clear how to solve the problem:  the values of "-k" are from the negative value of 

        p(-2) = {{{(-2)^3 - 2*(-2)^2 - 4*(-2)}}} = -8

    to  {{{p(x)[max_]}}}  over the interval (-2,0).


Thus our task now is to find  {{{p(x)[max_]}}}  over the interval (-2,0).


It is easy: it is a standard typical Calculus problem.

We must take the derivative of p(x); equate it to zero; find the optimum value of x and calculate p(x) at this value of x.


So, the derivative of p(x) is  p'(x) = 3x^2 - 4x - 4;  the root  of p'(x) = 0 is the root of the equation

    3x^2 - 4x - 4 = 0,

and over the interval (-2,0) it is  x = {{{x[max]}}} = {{{(4 - sqrt(4^2 + 4*3*4))/6}}} = {{{(4 - sqrt(64))/6}}} = {{{(4 - 8)/6}}} = {{{(-4)/6}}} = {{{-2/3}}}.


Finally,  {{{p(x)[max_]}}} = {{{p(-2/3)}}} = {{{(-2/3)^3 - 2*(-2/3)^2 - 4*(-2/3)}}} = 1.481 (rounded).


Thus we found out that the values {-k} belong to the interval (-8,1.481).


It means that the values of "k" we are seeking for are in the interval  (-1.481,8).    <U>ANSWER</U>
</pre>


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