Question

How do you divide using synthetic division: `(2u^4 - 5u^3 - 12u^2 + 2u - 8)/(u - 4)`?

Answer 1

`(2u^4-5u^3-12u^2+2u-8)/(u-4)=color(red)(2u^3+3u^2+2)`
with a Remainder of `color(blue)0`
`color(white)("XXX")`(see below for solution method using synthetic division)

Explanation:

`{: ([0],,,color(grey)(u^4),color(grey)(u^3),color(grey)(u^2),color(grey)(u^1),color(grey)(u^0)), ([1],," | ",2,-5,-12,+2,-8), ([2],ul(+color(white)("xxx"))," | ",ul(color(white)(0)),ul(+8),ul(+12),ul(+0),ul(+8)), ([3],xxcolor(magenta)4," | ",color(red)2,color(red)(+3),color(white)(+0)color(red)(0),color(white)("+")color(red)(2),color(white)("+")0), ([4],,,color(grey)(u^3),color(grey)(u^2),color(white)(+0)color(grey)(u^1),color(white)("+")color(grey)(u^0)color(white)("+"),color(blue)("R")) :}`

Rows [0] and [4] are not really part of the synthetic division; they are here for reference purposes only.

Row [1] are the coefficients of the variables in row [0]

Values in Row [3] are the sum of the values in the same column from Rows [1] and [2]

Values in Row [2] are the product of the values from the previous column of Row [3] and `color(magenta)4` where `color(magenta)4` is the value of `u` necessary to make the divisor `(u-4)` equal to `0`

Answer 2

The remainder is `color(red)(0)` and the quotient is `=2u^3+3u^2+2`

Explanation:

Let's perform the synthetic division

`color(white)(aa)` `4` `color(white)(aaaaa)` `|` `color(white)(aaa)` `2` `color(white)(aaaaa)` `-5` `color(white)(aaaaaa)` `-12` `color(white)(aaaaa)` `2` `color(white)(aaaaaa)` `-8`
`color(white)(aaaaaaaaaaaa)` `------------`

`color(white)(aaaa)` `color(white)(aaaa)` `|` `color(white)(aaaa)` `color(white)(aaaaaaa)` `8` `color(white)(aaaaaaa)` `12` `color(white)(aaaaa)` `0` `color(white)(aaaaaaaa)` `8`
`color(white)(aaaaaaaaaaaa)` `------------`

`color(white)(aaaa)` `color(white)(aaaa)` `|` `color(white)(aaa)` `2` `color(white)(aaaaaaa)` `3` `color(white)(aaaaaaaa)` `0` `color(white)(aaaaa)` `2` `color(white)(aaaaaaaa)` `color(red)(0)`

The remainder is `color(red)(0)` and the quotient is `=2u^3+3u^2+0u+2`

`(2u^4-5u^3-12u^2+2u-8)/(u-4)=2u^3+3u^2+0u+2`