Question

How do you use synthetic division to divide`(x^2+4x+1)` by `x-5`?

Answer 1

`x+9+46/(x-5)`

For explanation, see below.

Explanation:

Dividing by `x-5` means that the corresponding root of `5` will be in the upper left of the synthetic division. The numbers up top will be the coefficients of `x^2+4x+1`, which is `1,4,1`.

`{:(ul5"|"," 1"," 4"," 1"," "),(ul" ",ul" ",ul" ",ul" ",+),(" "," "," ","| "," "):}`

Move the `1` in the first column down to the bottom.

`{:(ul5"|"," 1"," 4"," 1"," "),(ul" ",ul" ",ul" ",ul" ",+),(" "," 1"," ","| "," "):}`

Multiply the `1` by the `5` in the corner for a product of `5`, which goes in the next column.

`{:(ul5"|"," 1"," "4," 1"," "),(ul" ",ul" ",ul(" "color(red)5),ul" ",+),(" ","      "color(red)1," ","| "," "):}`

Add the `4` and the `5` in the new column.

`{:(ul5"|"," 1"," "4," 1"," "),(ul" ",ul" ",ul(" "color(red)5),ul" ",+),(" ","      "color(red)1," "9,"| "," "):}`

Multiply the `9` by the `5` in the corner for `45`, which goes in the next column.

`{:(ul5"|"," 1"," "4," 1"," "),(ul" ",ul" ",ul(" "color(red)5),ul("    "color(blue)(45)),+),(" ","      "color(red)1," "color(blue)9,"| "," "):}`

Add the `1` and `45`. This goes in the "remainder" spot.

`{:(ul5"|"," 1"," "4," 1"," "),(ul" ",ul" ",ul(" "color(red)5),ul("    "color(blue)(45)),+),(" ","      "color(red)1," "color(blue)9,"|46"," "):}`

The numbers `1,9` in the bottom row correlate to the coefficients of the quotient. The `46` is the numerator of the remainder quotient:

`(x^2+4x+1)/(x-5)=x+9+46/(x-5)`