The prior lesson introduced long division of polynomials.
Using long division, a ‘new name’ is obtained for a fraction of polynomials:
$$
\cssId{s3}{\frac{N(x)}{D(x)}}
\cssId{s4}{= Q(x) + \frac{R(x)}{D(x)}}$$
For example, long division renames
$\displaystyle\frac{\overbrace{x^38x+2}^{N(x)}}{\underbrace{x+3}_{D(x)}}$
as
$\displaystyle\overbrace{x^2  3x + 1}^{Q(x)} + \frac{\overbrace{\ \ 1\ \ }^{R(x)}}{\underbrace{\ x+3\ }_{D(x)}}$ .
A quickandeasy 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.
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 $\,P(x)\,.$
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 $$ \cssId{s39}{\frac{N(x)}{D(x)} = Q(x) + \frac{R(x)}{D(x)}} $$ or, equivalently, $$ \cssId{s41}{N(x) = D(x)Q(x) + R(x)} $$ where either



$$\cssId{s67}{\frac{x^3  x^2 + x  1}{x1} = x^2 + 1}$$  $$\cssId{s68}{\frac{x^3  x^2 + x  1}{x^2+1} = x  1}$$ 
or, equivalently, $$\cssId{s70}{x^3  x^2 + x  1 = (x1)(x^2 + 1)}$$ 
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
