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  4. Find the product of uncommon real roots of the two polynomials P(x)=x4+2 x3-8 x2-6 x+15 and O(x)=x3+4 x2-x-10

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Q. Find the product of uncommon real roots of the two polynomials $P(x)=x^4+2 x^3-8 x^2-6 x+15$ and $O(x)=x^3+4 x^2-x-10$

Complex Numbers and Quadratic Equations

Solution:

$ x=-2$ is a root of the second polynomial $Q(x)$. So $Q(x)=(x+2)\left(x^2+2 x-5\right)$ $x=-2$ is not a root of the first polynomial $P(x)$.
Then check if $\left(x^2+2 x-5\right)$ is the root of $P(x)$. $P(x)=\left(x^2-3\right)\left(x^2+2 x-5\right)$. So the product of uncommon real roots is $(-3)(-2)=6$