Question

How do you use polynomial synthetic division to divide `(3x^3-x+4)div(x-2/3)` and write the polynomial in the form `p(x)=d(x)q(x)+r(x)`?

Answer 1

`3x^3-x+4=(x-2/3)(3x^2+2x+1/3)+4 2/9`

Explanation:

To divide `3x^3+0x^2-x+4` by `x-2/3`

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

`color(white)(1)|color(white)(X)3" "color(white)(X)0color(white)(XX)-1" "" "4`
`color(white)(1)|" "color(white)(X)`
`" "stackrel("—————————————)`

Two Put `2/3` in the divisor at the left as `x-2/3=0` gives `x=2/3`

`2/3|color(white)(X)3" "color(white)(X)0color(white)(XX)-1" "" "4`
`color(white)(xx)|" "color(white)(X)`
`" "stackrel("—————————————)`

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

`2/3|color(white)(X)3" "color(white)(X)0color(white)(XX)-1" "" "4`
`color(white)(X)|" "color(white)(X)`
`" "stackrel("—————————————)`
`color(white)(X)|color(white)(X)color(blue)3`

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

`2/3|color(white)(xx)3" "color(white)(X)0color(white)(XX)-1" "" "4`
`color(white)(xx)|" "color(white)(XXx)2`
`" "stackrel("—————————————)`
`color(white)(xx)|color(white)(X)color(blue)3`

Five Add down the column.

`2/3|color(white)(X)3" "color(white)(X)0color(white)(XX)-1" "" "4`
`color(white)(X)|color(white)(XXXX)2`
`color(white)(1)stackrel("—————————————)`
`color(white)(X)|color(white)(X)color(blue)3color(white)(XXx)color(red)2`

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

`2/3|color(white)(X)3" "color(white)(X)0color(white)(XX)-1" "" "4`
`color(white)(X)|" "color(white)(XxX)2color(white)(XXX)4/3color(white)(XxX)2/9`
`color(white)(1)stackrel("—————————————)`
`color(white)(X)|color(white)(X)color(blue)3color(white)(XxX)color(red)2color(white)(XXX)color(red)(1/3)color(white)(XxX)color(red)(4 2/9)`

Hence remainder is `4 2/9` and quotient is `3x^2+2x-1/3`

and hence `3x^3-x+4=(x-2/3)(3x^2+2x+1/3)+4 2/9`