Question

How do you divide `(x^4-8x^2+16)div(x+2)` using synthetic division?

Answer 1

`(x^4-8x^2+16) div (x+2) = color(red)(1x^3-2x^2-4x+8)`
with a Remainder of `color(green)(0)`

Explanation:

`x^4-8x^2+16 =color(blue)1x^4color(blue)(+0)x^3color(blue)(-8)x^2color(blue)(+0)x^1color(blue)(+16)x^0`

To divide by `(x+2)` we perform synthetic substitution with `x=color(magenta)(""(-2))`

#{: (,color(white)("xxx"),color(grey)(x^4),color(grey)(x^3),color(grey)(x^2),color(grey)(x^1),color(grey)(x^0),color(white)("xxx"),"row 0"), (,,color(blue)1,color(blue)(+0),color(blue)(-8),color(blue)(+0),color(blue)(+16),,"row 1"), (ul(color(white)("xx")+),,ul(color(white)("xxx")),ul(-2),ul(+4),ul(+8),ul(-16),,"row 2"), (xxcolor(magenta)(""(-2)),,color(red)(1),color(red)(-2),color(red)(-4),color(red)(+8),color(white)("xx")color(green)0,,"row 4"), (,,color(grey)(x^3),color(grey)(x^2),color(grey)(x^1),color(grey)(x^0),color(grey)("Rem."),,) :}#

The values for each column for row 4 are the sum of the values in rows 2 and 3 for that column.

The values for each column of row 3 are the product of `color(magenta)(""(-2))` and the value in row 4 of the previous column.

Answer 2

`color(magenta)(x^3-2x^2-4x+8` (using long division)

Explanation:

`(x^4-8x^2+16)-:(x+2)`

`color(white)(..........)color(white)(.)color(magenta)(x^3-2x^2-4x+8`
`x+2|overline(x^4+0x^3-8x^2+0x-16)`
`color(white)(............)ul(x^4+2x^3)`
`color(white)(..............)-2x^3-8x^2`
`color(white)(................)ul(-2x^3-4x^2)`
`color(white)(.........................)-4x^2+0x`
`color(white)(...........................)ul(-4x^2-8x)`
`color(white)(.......................................)8x-16`
`color(white)(.......................................)ul(8x-16)`