Question 305472
<font face="Garamond" size="+2">

Provide the missing terms:

*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 4x^3\ +\ 0x^2\ +\ 0x\ +\ 7]

*[tex \Large x] goes into *[tex \Large 4x^3\ \ \ ],*[tex \Large 4x^2] times.  Write *[tex \Large 4x^2] over the *[tex \Large 0x^2].

*[tex \Large 4x^2] times *[tex \Large -2] is *[tex \Large -8x^2].  Write *[tex \Large -8x^2] under the *[tex \Large 0x^2].  *[tex \Large 4x^2] times *[tex \Large x] is *[tex \Large 4x^3].  Write *[tex \Large 4x^3] under the *[tex \Large 4x^3].

Subtract (Change the sign and Add) *[tex \Large 4x^3\ -\ 8x^2] from *[tex \Large 4x^3\ +\ 0x^2].  Equals *[tex \Large 8x^2].

Bring down the *[tex \Large 0x] to form *[tex \Large 8x^2\ +\ 0x].

*[tex \Large x] goes into *[tex \Large 8x^2\ \ \ ],*[tex \Large 8x] times.  Write *[tex \Large 8x] over the *[tex \Large 0x].

*[tex \Large 8x] times *[tex \Large -2] is *[tex \Large -16x].  Write *[tex \Large -16x] under the *[tex \Large 0x].  *[tex \Large 8x] times *[tex \Large x]
is *[tex \Large 8x^2].  Write *[tex \Large 8x^2] under the *[tex \Large 8x^2].

Subtract (Change the sign and add) the *[tex \Large 8x^2\ -\ 16x] from  the *[tex \Large 8x^2\ -\ 0x]. Equals *[tex \Large 16x]

Bring down the *[tex \Large 7] to form *[tex \Large 16x\ +\ 7].

*[tex \Large x] goes into *[tex \Large 16x\ \ \ ],*[tex \Large 16] times.  Write *[tex \Large 16] over the *[tex \Large 7].

*[tex \Large 16] times *[tex \Large -2] is *[tex \Large -32].  Write *[tex \Large -32] under the *[tex \Large 7].  *[tex \Large 16] times *[tex \Large x]
is *[tex \Large 16x].  Write *[tex \Large 16x] under the *[tex \Large 16x].

Subtract (Change the sign and add) the *[tex \Large 16x\ -\ 32] from  the *[tex \Large 16x\ +\ 7]. Equals *[tex \Large 39]

Quotient is *[tex \Large 4x^2\ +\ 8x\ +\ 16]

Remainder is *[tex \Large 39]

Hence:

*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \frac{4x^3\ +\ 7}{x\ -\ 2}\ =\ 4x^2\ +\ 8x\ +\ 16\ +\ \frac{39}{x\ -\ 2}]

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