1.6 Factorizing of real polynomials

Thus one can express $g(x)$ as a product of:

(i) linear factors $(x-a)$, where $g(a)=0$; and

(ii) quadratics $u=x^{2}+2bx+c$ $(b^{2}<c)$ with no real roots.

Unfortunately, there is in general no easy way to find the roots, so we often need to rely on guesswork. Thus we factorize

where $a_{1},\dots,a_{s}$ are the real roots of $g(x)$ and $q_{1}(x),\dots,q_{j}(x)$ are irreducible quadratics. Now we gather together the equal factors, and write $g(x)$ as a product of powers of factors:

$\bullet$$Q(x)^{n}$, where $Q(x)$ is an irreducible quadratic;

$\bullet$$(x-b)^{m}$, where $g(b)=0$.

If $m=2$ then we call $b$ a double root; if $m=3$ then $b$ is a triple root.