Question

How do you use synthetic division to show that x=2 is a zero of `x^3-7x+6=0`?

Answer 1

Please see below.

Explanation:

To divide `x^3-7x+6=x^3+0x^2-7x+6` by `x-2` (as we have to show `x=2`, a zero).

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)-7" "" "6`
`color(white)(1)|" "color(white)(X)`
`" "stackrel("—————————————)`

Two As `x=2` we put `2` at the left.

`2|color(white)(x)1" "color(white)(X)0color(white)(XX)-7" "" "6`
`color(white)(x)|" "color(white)(XX)`
`" "stackrel("—————————————)`

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

`2|color(white)(x)1" "color(white)(X)0color(white)(XX)-7" "" "6`
`color(white)(x)|" "color(white)(X)`
`" "stackrel("—————————————)`
`color(white)(x)|color(white)(x)color(red)1`

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

`2|color(white)(x)1" "color(white)(X)0color(white)(XX)-7" "" "6`
`color(white)(x)|" "color(white)(XX)2`
`" "stackrel("—————————————)`
`color(white)(x)|color(white)(x)color(blue)1`

Five Add down the column.

`2|color(white)(x)1" "color(white)(x)0color(white)(XX)-7" "" "6`
`color(white)(x)|" "color(white)(XX)2`
`" "stackrel("—————————————)`
`color(white)(x)|color(white)(x)color(blue)1color(white)(X11)color(red)2`

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

`2|color(white)(x)1" "color(white)(X)0color(white)(XX)-7" "" "6`
`color(white)(x)|" "color(white)(XX)2color(white)(xxxxx)4color(white)(X)-6`
`" "stackrel("—————————————)`
`color(white)(x)|color(white)(x)color(blue)1color(white)(X11)color(red)2color(white)(XX)color(red)-3color(white)(XXX)color(red)0`

Hence, Quotient is `x^2+2x-1` and remainder is `0`.

As remainder is `0`, `x=2` is a zero of `x^3-7x+6`.