Given two polynomials $p_1,p_2\in\Bbb{F}_p[x]$, how to define polynomial $p_3$ that has all roots of $p_1$ except those that are $p_2$'s roots?

Question

Hypothesis: all polynomials are define over field $\mathbb{F}_p$, where $p$ is a large prime number.

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Consider we have two polynomials, $p_1(x)$ and $p_2(x)$ (as defined above).

For simplicity assume they have some roots in common.

Question: How can we define a polynomial $p_3(x)$ (using $p_1(x)$ and $p_2(x)$) that contains those roots of $p_1(x)$ that are not roots of $p_2(x)$ ?

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To be more clear, consider example below:

$p_1(x)=(x-a)\cdot(x-b)$

$p_2(x)=(x-a)\cdot(x-c)$

I need to know how to define $p_3(x)=x-b$, using polynomials $p_1(x)$ and $p_2(x)$

Answer 1

If you can compute the greatest common divisor of two polynomials, then $$p_3(x)=\frac{p_1(x)}{\gcd(p_1(x),p_2(x))},$$ will do the trick. This is always possible over (finite) fields.

Answer 2

HINT:

THe formula given above works say if the roots of $p_1(x)$ are simple ( multiplicity $1$) but may fail otherwise. For example, take $p_1(x)= x^7$, $p_2(x) = x^2$. Hence, we need to eliminate the common roots of $p_1(x)$ and $p_2(x)$ up to the maximum multiplicity of those common roots appearing in $p_1(x)$. To do this we could calculate succesively \begin{eqnarray} q_0(x)& =& p_1(x) \\ q_1(x)& =& \frac{q_0(x)}{ \gcd( q_0(x), p_2(x))}\\ \ldots \\ q_{m}(x)& = &\frac{q_{m-1}(x)}{ \gcd( q_{m-1}(x), p_2(x)) } \end{eqnarray} till the sequence $g_{m}(x)$ stabilizes, that is, till $\gcd (q_m(x), p_2(x) ) = 1$, and take that last $q_m(x)$ as the desired polynomial.