Binary string to boolean polynomial

Question

I have to find the corresponding Boolean polynomial to the binary string $1101$ $ 1001$.

To start, I think that I have to use a truth table to find the values for $x_1$, $x_2$, and $x_3$.

We get:

$0=0000$
$x_1=00001111$
$x_2=00110011$
$x_3=01010101$
and so:
$x_1+x_2=00111100$
$x_1+x_3=01011010$
$x_2+x_3=01100110$
$1=11111111$
$1+x_1=11110000$
$1+x_2=11001100$
$1+x_3=10101010$
$1+x_1+x_2=11000011$
$1+x_1+x_3=10100101$
$1+x_2+x_3=10011001$
$1+x_1+x_2+x_3=10010110$

I think that I have to find a linear combination of the above in order to get the desired binary string, however I'm not sure that this is the correct process. Any hints would be greatly appreciated!

Also does the space between the binary string I have to convert change the polynomials I should be using?

Answer 1

\begin{eqnarray*} 1 &=& 11111111 \\ x_2 &=& 00110011 \\ x_2x_3 &=& 00010001 \\ x_1 x_3 &=& 00000101 \\ x_1 x_2 x_3 &=& 00000001 \\ \end{eqnarray*}