Question

How do you use polynomial synthetic division to divide `(3x^2-2x+1) div(x-1)` and write the polynomial in the form `p(x)=d(x)q(x)+r(x)`?

Answer 1

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

Explanation:

To divide `(3x^2-2x+1)` by `(x-1)` using synthetic division, we should take following steps

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)-2color(white)(XXXX)1"`
`color(white)(1)|" "color(white)(X)`
`" "stackrel("—————————————)`

Two Put the divisor at the left, but for this put `x-1=0`, which gives `x=1`

`1|color(white)(X)3" "color(white)(X)-2color(white)(XXXX)1`
`color(white)(1)|" "color(white)(X)`
`" "stackrel("—————————————)`

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

`1|color(white)(X)3" "color(white)(X)-2color(white)(XXXX)1`
`color(white)(1)|" "color(white)(X)`
`" "stackrel("—————————————)`
`color(white)(1)|color(white)(X)color(blue)3`

Four Multiply the result of `1xxcolor(blue)3=3`, here `1` comes from divisor, and put the product in the next column.

`1|color(white)(X)3" "color(white)(X)-2color(white)(XXXX)1`
`color(white)(1)|" "color(white)(XXXX)3`
`" "stackrel("—————————————)`
`color(white)(1)|color(white)(X)color(blue)3`

Five Add down the column.

`1|color(white)(X)3" "color(white)(X)-2color(white)(XXXX)1`
`color(white)(1)|" "color(white)(XXXX)3`
`" "stackrel("—————————————)`
`color(white)(1)|color(white)(X)color(blue)3color(white)(XXXx)color(blue)1`

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

`1|color(white)(X)3" "color(white)(X)-2color(white)(XXXX)1`
`color(white)(1)|" "color(white)(XXXX)3color(white)(XXXX)1`
`" "stackrel("—————————————)`
`color(white)(1)|color(white)(X)color(blue)3color(white)(XXXx)color(blue)1color(white)(XXXX)color(red)2`

WE find that quotient is `(x+1)` and remainder is `2`.

Hence `(3x^2-2x+1)=color(blue)((3x+1))(x-1)+color(red)2`