Question

How do you divide `(x^3+x^2+3x+1)/(x-3)`?

Answer 1

`x^2 +4x+15 " rem " 46/(x-3)`

Explanation:

Division by synthetic division is the simplest and quickest method in this case. We will only use the numerical coefficients.

`x-3 =0 " "rarr x =3" "` goes outside on the left.

`(1x^3 +1x^2 +3x+1)/(x-3)`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`" "3|" "1" "1" "3" "1`
`" "|ul" "darrul" "`
`" "1color(white)(xxxxxxxxxxxxxxx)larr` bring down the `1`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

`" "color(red)(3)|" "1" "1" "3" "1`
`" "|ul" "darrul" "color(blue)(3)ulcolor(white)(xxxxxxxxxxx)larrcolor(red)(3xx1) = color(blue)(3)`
`" "color(red)(1)" "color(blue)(4)color(white)(xxxxxxxxxxx)larr 1+color(blue)(3 =4)`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

`" "color(red)(3)|" "1" "1" "3" "1`
`" "|ul" "darrul" "3ul" "color(green)(12)ulcolor(white)(xxxxxxxxx)larrcolor(red)(3xxcolor(blue)(4) color(green)(=12)`
`" "1" "color(blue)(4)" "color(green)(15)color(white)(xxxxxxxxx)larr 3+color(green)(12 =15)`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

`" "color(red)(3)|" "1" "1" "3" "1`
`" "|ul" "darrul" "3ul" "12" "color(magenta)(45)ulcolor(white)(xxxxxx)larrcolor(red)(3xxcolor(green)(15) color(magenta)(=45)`
`" "1" "color(blue)(4)" "color(green)(15)" "color(magenta)(46)color(white)(xxxxxx)larr 1+color(magenta)(45 =46)`
`color(white)(xxxxxxxxxxxxxxxx)uarr`
`color(white)(xxxxxxxxxxxxxxx)"remainder"`

The bottom line gives the numerical coefficients of the quotient.
`x^3 divx =x^2`

Quotient:

`1x^2 +color(blue)(4)x +color(green)15" rem " color(magenta)(46)`

Answer 2

The remainder is `46` and the quotient is `=x^2+4x+15`

Explanation:

Let's perform the synthetic division

`color(white)(aaaa)` `3` `color(white)(aaaa)` `|` `color(white)(aaaa)` `1` `color(white)(aaaa)` `1` `color(white)(aaaaaa)` `3` `color(white)(aaaaaaa)` `1`
`color(white)(aaaaaaaaaaaa)`_________

`color(white)(aaaa)` `color(white)(aaaaaaa)` `|` `color(white)(aaaa)` `color(white)(aaa)` `3` `color(white)(aaaaa)` `12` `color(white)(aaaaaa)` `45`
`color(white)(aaaaaaaaaaaa)`________

`color(white)(aaaa)` `color(white)(aaaaaaa)` `|` `color(white)(aaa)` `1` `color(white)(aaa)` `4` `color(white)(aaaaa)` `15` `color(white)(aaaaaa)` `color(red)(46)`

The remainder is `46` and the quotient is `=x^2+4x+15`

`(x^3+3x^2+3x+1)/(x-3)=x^2+4x+15+46/(x-3)`