Question 182573
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*[tex \LARGE \text{          }\math \frac{x^3-2x^2+x-2}{x-2}]


Before you do anything else, determine values of the variable that must be excluded.  For a rational expression, set the denominator equal to zero and solve:


*[tex \LARGE \text{          }\math x - 2 = 0 \ \ \Rightarrow\ \ x = 2]


So, regardless of what the expression looks like after simplification, the value 2 must remain excluded.


Use Polynomial Long Division:


*[tex \Large x - 2 \text{  }\math \overline{|x^3\ -2x^2\ +x\ -2}]


*[tex \Large x] goes into *[tex \Large x^3,\  x^2] times, so:


*[tex \Large \text{              }\math x^2]
*[tex \Large x - 2 \text{  }\math \overline{|x^3\ -2x^2\ +x\ -2}]


*[tex \Large x^2 \times x - 2 = x^3 - 2x^2] so


*[tex \Large \text{              }\math x^2]
*[tex \Large x - 2 \text{  }\math \overline{|x^3\ -2x^2\ +x\ -2}]
*[tex \Large \text{       }\math \underline{x^3\  - 2x^2]


Change signs of subtrahend and add:


*[tex \Large \text{              }\math x^2]
*[tex \Large x - 2 \text{  }\math \overline{|x^3\ -2x^2\ +x\ -2}]
*[tex \Large \text{       }\math \underline{x^3\  - 2x^2]
*[tex \Large \text{       }\math \ \ \ \ \ \ \ 0]


Bring down the *[tex \Large x]


*[tex \Large \text{              }\math x^2\ \ \ \]
*[tex \Large x - 2 \text{  }\math \overline{|x^3\ -2x^2\ +x\ -2}]
*[tex \Large \text{       }\math \underline{x^3\  - 2x^2]
*[tex \Large \text{       }\math \ \ \ \ \ \ \ 0\ \ \,+x]


*[tex \Large x] goes into *[tex \Large 0,\  0] times, so:


*[tex \Large \text{              }\math x^2\ \ \ \, 0]
*[tex \Large x - 2 \text{  }\math \overline{|x^3\ -2x^2\ +x\ -2}]
*[tex \Large \text{       }\math \underline{x^3\  - 2x^2]
*[tex \Large \text{       }\math \ \ \ \ \ \ \ 0\ \ \,+x]


*[tex \Large 0 \times x - 2 = 0] so


*[tex \Large \text{              }\math x^2\ \ \ \, 0]
*[tex \Large x - 2 \text{  }\math \overline{|x^3\ -2x^2\ +x\ -2}]
*[tex \Large \text{       }\math \underline{x^3\  - 2x^2]
*[tex \Large \text{       }\math \ \ \ \ \ \ \ 0\ \ \,+x]
*[tex \Large \text{       }\math \ \ \ \ \ \ \ \underline{\ \ \ \ \ \ \,0}]


Change signs of subtrahend and add:


*[tex \Large \text{              }\math x^2\ \ \ \, 0]
*[tex \Large x - 2 \text{  }\math \overline{|x^3\ -2x^2\ +x\ -2}]
*[tex \Large \text{       }\math \underline{x^3\  - 2x^2]
*[tex \Large \text{       }\math \ \ \ \ \ \ \ 0\ \ \,+x]
*[tex \Large \text{       }\math \ \ \ \ \ \ \ \underline{\ \ \ \ \ \ \,0}]
*[tex \Large \text{       }\math \ \ \ \ \ \ \ \ \ \ \ \ \ x]


Bring down the -2:


*[tex \Large \text{              }\math x^2\ \ \ \, 0]
*[tex \Large x - 2 \text{  }\math \overline{|x^3\ -2x^2\ +x\ -2}]
*[tex \Large \text{       }\math \underline{x^3\  - 2x^2]
*[tex \Large \text{       }\math \ \ \ \ \ \ \ 0\ \ \,+x]
*[tex \Large \text{       }\math \ \ \ \ \ \ \ \underline{\ \ \ \ \ \ \,0}]
*[tex \Large \text{       }\math \ \ \ \ \ \ \ \ \ \ \ \ \ x\ -2]


*[tex \Large x] goes into *[tex \Large x,\  1] times, so:


*[tex \Large \text{              }\math x^2\ \ \ \, 0\ +1]
*[tex \Large x - 2 \text{  }\math \overline{|x^3\ -2x^2\ +x\ -2}]
*[tex \Large \text{       }\math \underline{x^3\  - 2x^2]
*[tex \Large \text{       }\math \ \ \ \ \ \ \ 0\ \ \,+x]
*[tex \Large \text{       }\math \ \ \ \ \ \ \ \underline{\ \ \ \ \ \ \,0}]
*[tex \Large \text{       }\math \ \ \ \ \ \ \ \ \ \ \ \ \ x\ -2]


Finally:

*[tex \Large \text{              }\math x^2\ \ \ \, 0\ +1]
*[tex \Large x - 2 \text{  }\math \overline{|x^3\ -2x^2\ +x\ -2}]
*[tex \Large \text{       }\math \underline{x^3\  - 2x^2]
*[tex \Large \text{       }\math \ \ \ \ \ \ \ 0\ \ \,+x]
*[tex \Large \text{       }\math \ \ \ \ \ \ \ \underline{\ \ \ \ \ \ \,0}]
*[tex \Large \text{       }\math \ \ \ \ \ \ \ \ \ \ \ \ \ x\ -2]
*[tex \Large \text{       }\math \ \ \ \ \ \ \ \ \ \ \ \ \underline{\ x\ -2}]
*[tex \Large \text{       }\math \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \,0]


Since the remainder is zero, *[tex \Large x - 2] divides *[tex \Large x^3 -2x^2 + x =2] evenly.  Therefore we can say:


*[tex \LARGE \text{          }\math \frac{x^3-2x^2+x-2}{x-2} = x^2 + 1\ \ \Leftrightarrow\ \ x \neq 2]


John
*[tex \LARGE e^{i\pi} + 1 = 0]
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