Question

How do you use synthetic division to divide `(x^3-729)/(x-9)`?

Answer 1

Quotient is `x^2+9x+81` and remainder is `0`.

Explanation:

To divide `x^3-729` or `x^3+0x^2+0x-729` by `x+2`

One Write the coefficients of `x` in the dividend inside an upside-down division symbol.

`color(white)(1)|color(white)(X)1" "color(white)(X)0color(white)(XX)0" "" "-729`
`color(white)(1)|" "color(white)(X)`
`" "stackrel("—————————————)`

Two As `x-9=0` gives `x=9` put `9` at the left.

`9color(white)(X)|1" "color(white)(X)0color(white)(XX)0" "" "-729`
`color(white)(xx)|" "color(white)(XX)`
`" "stackrel("—————————————)`

Three Drop the first coefficient of the dividend below the division symbol.

`9color(white)(X)|1" "color(white)(X)0color(white)(XX)0" "" "-729`
`color(white)(xx)|" "color(white)(X)`
`" "stackrel("—————————————)`
`color(white)(xx)|color(white)()color(red)1`

Four Multiply the result by the constant, and put the product in the next column.

`9color(white)(X)|1" "color(white)(X)0color(white)(XX)0" "" "-729`
`color(white)(xx)|" "color(white)(Xx)9`
`" "stackrel("—————————————)`
`color(white)(xx)|color(white)()color(blue)1`

Five Add down the column.

`9color(white)(X)|1" "color(white)(X)0color(white)(XX)0" "" "-729`
`color(white)(xx)|" "color(white)(Xx)9`
`" "stackrel("—————————————)`
`color(white)(xx)|color(white)()color(blue)1color(white)(xxx)color(red)9`

Six Repeat Steps Four and Five until you can go no farther.

`9color(white)(X)|1" "color(white)(X)0color(white)(XX)0" "" "-729`
`color(white)(xx)|" "color(white)(Xx)9color(white)(XX)81color(white)(XXxx)729`
`" "stackrel("—————————————)`
`color(white)(xx)|color(white)()color(blue)1color(white)(xxx)color(red)9color(white)(XX)color(red)81color(white)(XXXXX)color(red)0`

Hence, Quotient is `x^2+9x+81` and remainder is `0`.