Long division of polynomials

One of the important facts about the algebra of polynomials is that the division algorithm works for polynomials. In fact, division with remainder is somewhat easier for polynomials than it is for numbers, since there is no guessing and carrying involved. The theorem states that

Theorem 1   If $a(x)$ and $b(x)\neq 0$ are polynomials over the field $F$ then there are polynomials $q(x)$ and $r(x)$ which are unique up to a factor in $F$, such that $a(x)=b(x)q(x)+r(x)$ and the degree of $r(x)$ is less than the degree of $b(x)$

The algorithm looks just like long division of numbers, only using powers of $x$ as place holders instead of powers of 10:

$$\begin{array}{cccccccc}&&&&&&2x& +6 \\ \cline{4-8} x^2 & -3 x... ...&&& 6x^2 & -18x & +6 \\ \cline{6-8} &&&&&& 22x &-4 \end{array}$$

So $2x^3+6x+2= (x^2-3x+1)(2x+6) +(22x-4)$.