We would like to understand polynomials by understanding what factors they are
made of. Remember that the zeros of a polynomial \(p(x)\) are the values of \(x\)
for which \(p(x)=0\). Remember also that the zeros of a polynomial can be real
numbers or complex numbers.
The factors and the zeros (real and complex) of a polynomial are directly
related. Every zero creates a factor and every factor describes a zero. In algebra
this is known as the Factor Theorem.
A theorem in mathematics is a statement that may or may not be obvious but is
always true. You can demonstrate that the Factor Theorem is true by learning how
to divide polynomials with the division algorithm.
Any two polynomials can be divided the same way that any two numbers can be divided
with long division. Of course, dividing by zero, whether in the case of numbers or
polynomials is still strictly forbidden.
Synthetic division is efficient way to divide a polynomial by a linear factor or
evaluate a polynomial function at a given number, real or complex.
Armed with synthetic division and two other fundamental truths, the Rational Zeros
Theorem and Descartes' Rule of signs, we can completely factor a polynomial using
it's real and complex zeros. This is a significant accomplishment and it may
require a significant amount of calculating.
A side benefit of being able to completely factor polynomials is that we can draw
complete and detailed graphs of rational functions \(p(x)/q(x)\), creating a
framework for the graph of a rational function with the real zeros of the
numerator \(p(x)\) and denominator \(q(x)\).