<PRE><B>Question 1123634</B>
&lt;br&gt;
Since the problem asks us to find all the zeros using synthetic division, we can assume that all the roots are rational numbers.&lt;br&gt;
The rational roots theorem tells us the possible rational roots are of the form plus or minus p/q, where p is a factor of the constant term (24) and q is a factor of the leading coefficient (6).  That gives us a long list of possible roots:&lt;br&gt;
+ or - {1, 2, 3, 4, 6, 8, 12, 24, 1/2, 3/2, 1/3, 2/3, 4/3, 8/3, 1/6}&lt;br&gt;
(and I might have missed some....)&lt;br&gt;
Before we start our search, let&#39;s see what we know about the roots.&lt;br&gt;
Descartes&#39; rule of signs tells us that the number of positive roots is either 2 or 0 and the number of negative roots is either 2 or 0.  Since there are 4 roots, we then know that 2 are positive and 2 are negative.&lt;br&gt;
We also know that the sum of the roots is (-(-5))/6 = 5/6 and their product is 24/6 = 4.&lt;br&gt;
Now we are ready to start our search.  Trying fractional roots, either with synthetic division or by evaluating the polynomial, is difficult; so start with possible roots that are integers.&lt;br&gt;
For small integers, evaluating the polynomial is often faster and easier than using synthetic division.  So I personally would start my work by evaluating the polynomial for small integer values.&lt;br&gt;
1 and -1 are always the easiest to try.  A small bit of mental arithmetic shows that 1 is not a root but -1 is.  So now we use synthetic division to extract that root.&lt;br&gt;&lt;pre&gt;

   -1 | 6  -5  -25  10  24
      |    -6   11  14 -24
      --------------------
        6 -11  -14  24   0&lt;br&gt;&lt;/pre&gt;
The reduced polynomial is 6x^3-11x^2-14x+24.  Because mental arithmetic is easy for me, my preference at this point would be to try 2 and -2 as roots by evaluating the polynomial for those values.  However, switching to synthetic division might be the best path for most people.&lt;br&gt;
So try 2 as a root next:&lt;br&gt;&lt;pre&gt;

  2 | 6  -11  -14  24
    |     12    2 -24
    -----------------
      6    1  -12   0&lt;br&gt;&lt;/pre&gt;
Aha! 2 is a root!  And the reduced polynomial 6x^2-x-12 is quadratic; and I certainly would not use synthetic division to find the last two roots.&lt;br&gt;
6x^2+x-12 = (3x-4)(2x+3)&lt;br&gt;
The last two roots are 4/3 and -3/2.&lt;br&gt;
Answer:  The four roots are -1, 2, 4/3, and -3/2.&lt;br&gt;
It turned out we did not use the numbers of positive and negative roots, or the sum and product of the roots, to find the four roots.  However, if the instructions had not been to use synthetic division, we might have been able to use those to find the roots by a different path.
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