Question

How do you use synthetic division to divide `x^3 + 2x^2 - 2x + 24` by `x+4`?

Answer 1

`frac{x^3 + 2x^2 - 2x + 24}{x + 4} = x^2 - 2x + 6`

Explanation:

Divide `x^3` by `x + 4`
Quotient `= x^2`
Remainder `= x^3 + 2x^2 - 2x + 24 - x^3 - 4 x ^2`
Remainder `= - 2x^2 - 2x + 24`

Divide `-2x^2` by `x + 4`
Quotient `= -2x`
Remainder `= - 2x^2 - 2x + 24 + 2x^2 + 8x`
Remainder `= 6x + 24`

Divide `6x` by `x + 4`
Quotient `=6`
Remainder `= 0`

Answer 2

Synthetic division gives: `x^2-2x+6`

Process shown in detail

Explanation:

Given: `(x^3+2x^2-2x+24)/(x+4)`

Consider the denominator `x+4`

Set `x+4=0 =>color(red)(x=-4)`

Now we construct the coefficient manipulation.

`color(white)("dddd") x^3+2x^2-2x+24`
`color(white)("ddd")darrcolor(white)("d.d") darrcolor(white)("ddd")darrcolor(white)("ddd")darr`

`color(red)(-4)|bar(1color(white)("ddd")2color(white)("ddd")-2color(white)("ddd")24`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
bring down the first value of `color(magenta)(1)`

`color(red)(-4)|bar(1color(white)("ddd")2color(white)("ddd")-2color(white)("ddd")24`
`color(white)("d..d")ul(|color(magenta)(darr )`
`color(white)("dddd")color(magenta)(1)`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`-4xxcolor(magenta)(1)=color(blue)(-4)`
Place the `color(blue)(-4)` under the 2

`color(red)(-4)|bar(1color(white)("ddd")2color(white)("ddd")-2color(white)("ddd")24`
`color(white)("d..d")ul(|color(magenta)(darr )color(white)("d")color(blue)(-4) )`
`color(white)("dddd")color(magenta)(1)`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Add the 2 and the `color(blue)(-4)`

`color(red)(-4)|bar(1color(white)("ddd")2color(white)("ddd")-2color(white)("ddd")24`
`color(white)("d..d")ul(|color(magenta)(darr )color(white)("d")color(blue)(-4) )`
`color(white)("dddd")color(magenta)(1)color(white)("d")-2`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(red)(-4)xx" the new "-2=+8`
Place the `-8` under the `-2`

`color(red)(-4)|bar(1color(white)("dddd")2color(white)("ddd")-2color(white)("ddd")24`
`color(white)("d..d")ul(|darr color(white)("d")-4color(white)("ddddd")8 )`
`color(white)("dddd")1color(white)("dd")-2`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Add the `-2" and the "8`

`color(red)(-4)|bar(1color(white)("dddd")2color(white)("ddd")-2color(white)("ddd")24`
`color(white)("d..d")ul(|darr color(white)("d")-4color(white)("ddddd")8 )`
`color(white)("dddd")1color(white)("dd")-2color(white)("ddddd")6`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(red)(-4)xx "the new "6 = -24`
Place the -24 under the existing +24
Add and you get the 0 so it is an exact division. No remainder.

`color(red)(-4)|bar(1color(white)("dddd")2color(white)("ddd")-2color(white)("ddd")24`
`color(white)("d..d")ul(|darr color(white)("d")-4color(white)("ddddd")8color(white)("ddd")24 )`
`color(white)("dddd")1color(white)("dd")-2color(white)("ddddd")6color(white)("ddd")0`
`color(white)("ddd.")darrcolor(white)("ddd")darrcolor(white)("dddd")darr`

`color(white)("dddd")x^2color(white)("d")-2xcolor(white)("dd")+6`