Question 1208426
<pre>

If [x+(1/3)] or [x-(-1/3)] were a factor then the factorization would be

{{{3x^4 + x^3 - 3x + 1}}}{{{""=""}}}{{{(x+1/3)("?"x^3+"?"x^2+"?"x+"?")}}}

Then {{{x+1/3=0}}} would give a zero {{{x=-1/3}}} of {{{3x^4 + x^3 - 3x + 1}}}.

So to find out, we use synthetic division to see if the remainder, which is the
same value as when -1/3 is substituted for x, is zero.  That happens when the
last term on the bottom right of the synthetic division is 0.  So here goes the
synthetic division:

Be sure to remember that {{{3x^4 + x^3 - 3x + 1}}} has a missing term in
{{{x^2}}}.  So we must consider it as {{{3x^4 + x^3 + red(0x^2) - 3x + 1}}} 

-1/3 | 3  1  <font color="red"><b>0</b></font> -3  1
     |<u>   -1  0  0  1</u>
       3  0  0 -3  2

The number on the bottom right did not turn out to be zero.  It came out to be
2 instead, so, no [x + (1/3)] is not a factor of  {{{3x^4 + x^3 - 3x + 1}}}.

Edwin</pre>