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Let $P$ and $Q$ be polynomials of degree $2$ and $3$ respectively. If we know the roots of both $P$ and $Q$, is there an easier way of finding the roots of the product $PQ$? Do we really have to multiply them to get a degree $5$ polynomial and look for roots?

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If $P$ has degree $2$ and $Q$ has degree $3$ then their product has degree $5$, not $6$. – Gerry Myerson Dec 9 '12 at 11:33
yeah, my Bad. it is 5 – doniyor Dec 9 '12 at 11:36
up vote 1 down vote accepted

If you know $P$ has roots $\alpha,\beta$ and $Q$ has roots $\gamma,\delta,\epsilon$, then you can write $P(x)=a(x-\alpha)(x-\beta)$ and $Q(x)=b(x-\gamma)(x-\delta)(x-\epsilon)$, so multiplying these together gives you that it has precisely the roots of $P$ and $Q$.

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thanks peter. great – doniyor Dec 9 '12 at 11:18

If you have the product of 2 polynomial, say $P(x)$ and $Q(x)$, the set of roots of $R(x) = P(x)Q(x)$ is the union of the set of roots of $P(x)$ and of $Q(x)$.

You don't have to multiply them and calculate the roots of all $R(x)$, since you already know them if you know the roots of $P(x)$ and of $Q(x)$.

In fact, if $x_P$ is a root of $P(x)$ (e.g. $P(x_P) = 0$), then you have that:

$$ R(x_P) = P(x_P)Q(x_P) = 0~ Q(x_P) = 0$$

so $x_P$ is a root of $R(x)$. Similarly you can do this by saying that $x_Q$ is a root of $Q(x)$, and then $x_Q$ is also a root of $R(x)$.

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