It is second-order because the highest power of is (only
non-negative integer powers of are allowed in this context). The
polynomial is also
monic
because its leading coefficient, the
coefficient of , is . By the fundamental theorem of algebra
(discussed further in §2.4), there are exactly two
roots
(or
zeros) of any
second order polynomial. These roots may be real or complex (to be defined).
For now, let's assume they are both real and denote them by
and . Then we have
and , and we can write

This is the factored form of the monic polynomial .
(For a non-monic polynomial, we may simply divide all coefficients
by the first to make it monic, and this doesn't affect the zeros.)
Multiplying out the symbolic factored form gives

Comparing with the original polynomial, we find we must have

This is a system of two equations in two unknowns. Unfortunately, it is a
nonlinear system of two equations in two
unknowns.^{2.1} Nevertheless, because it is so small,
the equations are easily solved. In beginning algebra, we did them by
hand. However, nowadays we can use a software tool such as Matlab or
Octave to solve very large systems of linear equations.

The factored form of this simple example is

Note that polynomial factorization rewrites a monic th-order
polynomial as the product of first-order monic polynomials,
each of which contributes one zero (root) to the product. This
factoring business is often used when working with digital
filters [66].