## Algebraic Long Division 1

Given a polynomials, $P(x)$ and $S(x)$ where $\mbox{deg}(P) \geq \mbox{deg}(S)$, it is possible to express $P(x)$ as $$ P(x) = S(x)Q(x) + R(x) $$ Where $Q(x)$ and $R(x)$ are polynomials. We call $R(x)$ the remainder and $Q(x)$ the quotient

## Algebraic Long Division 1

## Algebraic Long Division 2

We shall separate out each iteration of the process:

In the first instance, $x^4/ x^2 = x^2$ so we have

$$x^4 +3x^3 +6x^2+0x+2 - (x^2)(x^2 + 4x + 2) = -x^3 + 4x^2 +0x +2$$

In the second instance, $-x^3 / x^2 = - x$ so we have

$$-x^3+4x^2+0x+2 - (-x)(x^2+4x+2) = 8x^2 +2x +2$$

In the third instance, $8x^2 /x^2 = 8$,so we have

$$8x^2 +2x +2 - (8)(x^2+4x+2) = -30x-14$$

The process now halts as the remainder has a degree strictly lower than the divisor.

The Quotient is $x^2 -x + 8$ and the Remainder is $-30x-14$

This process is normally tabulated, but the expanded view is sometimes helpful to elucidate the process.