Question 1138616
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The polynomial is   p(x) = 2x^4 + 15x^3 + 31x^2 + 20x + 4


The possible rational roots, based on the Rational Roots theorem, are


  (+/- 4/1) = +/- 4;   (+- 4/2) = +/- 2;  (+- 2/1) = +- 2;  (+- 2/2) = +- 1;  (+- 1/2).


The plot of the polynomial is shown in the Figure below.



    {{{graph( 330, 330, -5, 2, -5, 5,
          2x^4 + 15x^3 + 31x^2 + 20x + 4
)}}}


    Plot y = {{{2x^4 + 15x^3 + 31x^2 + 20x + 4}}}


From the plot, is is clear that 


    -2 is very possible candidate;  

    {{{-1/2}}} is a potential candidate; and 

     the root between -4 and -5 is an irrational number.


Immediate direct check/substitution proves that -2 is the root and {{{-1/2}}} is the root, too.


Then the original polynomial is divisible by the product  (x+2)*(2x+1),  and long division gives the quotient


    {{{p(x)/((x+2)*(2x+1))}}} = x^2 +5x +2.


Use the quadratic formula and find two remaining real irrational roots of the polynomial.
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The lesson to learn from my post is <U>THIS</U>:


<pre>
    The Rational Root theorem is a good tool, but the analysis becomes much more quicker and productive, 

    if you use graphic calculator or plotting tool to visualize a polynomial.

    By doing in this way, you will be able to cut off easily the dead branches.
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