Product of uncommon real roots?

Question

The product of uncommon real roots of the two polynomials

$p(x)=x^4+2x^3-8x^2-6x+15$

And

$q(x)=x^3+4x^2-x-10$

My attempt was to form an equation of form

$p(x)+\lambda q(x)=0$

which will satisfy their common roots but generates extra roots, so doesn't really help.

Answer 1

There are several algebraic methods to obtain the roots which are not common roots. Using the Euclidean algorithm one obtains that $gcd(p(x),q(x))=x^2+2x-5$. Dividing $p(x)$ and $q(x)$ by the gcd, we obtain the factorizations $$ p(x)=(x^2 + 2x - 5)(x^2 - 3),\; q(x)=(x^2 + 2x - 5)(x+2). $$ So $x=-2$ and $x=\pm\sqrt{3}$ are the real roots which are not common roots.