Answer
$x+\frac{1}{3}$ is not a factor of the given polynomial.
Work Step by Step
The given expression is:-
$(3x^4+x^3-3x+1)\div (x+\frac{1}{3})$
Rewrite as descending powers of $x$.
$(3x^4+x^3+0x^2-3x+1)\div (x+\frac{1}{3})$
The divisor is $x+\frac{1}{3}$, so the value of $c=-\frac{1}{3}$.
and on the right side the coefficients of dividend in descending powers of $x$.
Perform synthetic division to obtain:
$\begin{matrix}
&-- &-- &--&--& --&& \\
-\frac{1}{3}) &3&1&0&-3&1 & &\\
& &-1 &0 &0 &1& &\\
& -- & -- & --& -- &--&& \\
& 3 & 0& 0 &-3 & 2&&\\
\end{matrix}$
The remainder is not zero.
Hence, $x+\frac{1}{3}$ is not a factor of the given polynomial.
