THE DIVISION ALGORITHM

The prior web exercise introduced long division of polynomials.

Using long division, a ‘new name’ is obtained for a fraction of polynomials: [beautiful math coming... please be patient]$$\frac{N(x)}{D(x)} = Q(x) + \frac{R(x)}{D(x)}$$ For example, long division renames     [beautiful math coming... please be patient]$\displaystyle\frac{\overbrace{x^3-8x+2}^{N(x)}}{\underbrace{x+3}_{D(x)}}$     as     [beautiful math coming... please be patient]$\displaystyle\overbrace{x^2 - 3x + 1}^{Q(x)} + \frac{\overbrace{\ \ -1\ \ }^{R(x)}}{\underbrace{\ x+3\ }_{D(x)}}$   .

A quick-and-easy check gives some confidence that these two expressions are indeed the same:

substituting $\,x = 0\,$ into $\displaystyle \frac{N(x)}{D(x)}$ gives: $\displaystyle \frac{0^3 - 8\cdot 0 + 2}{0 + 3}$ $\displaystyle = \frac 23$
substituting $\,x = 0\,$ into $\displaystyle Q(x) + \frac{R(x)}{D(x)}$ gives: $\displaystyle 0^2 - 3\cdot 0 + 1 + \frac{-1}{0 + 3} = 1 -\frac 13$ $\displaystyle = \frac 23$

The new expression for the fraction that is obtained by long division is often better to work with than the original.

Terminology used in the Long Division Process

In the long division process, when   $\displaystyle\,\frac{N(x)}{D(x)}\,$   is renamed as   $\displaystyle\,Q(x) + \frac{R(x)}{D(x)}\,$:

The Division Algorithm (below) firms up details about division of polynomials.
Here, the notation   ‘$\,\text{deg}(P(x))\,$’   is used to denote the degree of a polynomial $\,P(x)\,$.

the Division Algorithm dividing a polynomial by a polynomial
Let $\,N(x)\,$ and $\,D(x)\,$ be polynomials, with $\,D(x)\ne 0\,$.

There exist unique polynomials $\,Q(x)\,$ (called the quotient) and $\,R(x)\,$ (called the remainder) such that $$ \frac{N(x)}{D(x)} = Q(x) + \frac{R(x)}{D(x)} $$ or, equivalently, $$ N(x) = D(x)Q(x) + R(x) $$ where either
  • $R(x) = 0$;   or
  • $\deg(R(x)) \lt \text{deg}(D(x))$

Notes about the Division Algorithm

Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Synthetic Division
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
PROBLEM TYPES:
1 2 3 4 5
AVAILABLE MASTERED IN PROGRESS

(MAX is 5; there are 5 different problem types.)