## Factoring a Polynomial

Remember ``factoring polynomials''? Consider the second-order polynomial

*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

*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

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

*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*[68].

**Next Section:**

The Quadratic Formula

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