Question

How do you calculate this with Horner's method `(8x^3-20x^2+2x-2)/(2x-1)`?

I only know the anwser but I don't know how you get there. The answer should be `Q= 4x^2-8x-3` and R=-5

Can you explain it step by step (and with formula's if necessary) ?

Answer 1

The steps are given below and answer is

`8x^3-20x^2+2x-2=(2x-1)(4x^2-8x-3)-5`

Explanation:

Horner's method is better known as Synthetic Division and hence, what it means here is dividing the polynomial `8x^3-20x^2+2x-2` by `2x-1`. The steps for this are as follows:

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

`color(white)(1)|color(white)(X)8" "color(white)(X)-20color(white)(XX)2" "" "-2`
`color(white)(1)|" "color(white)(X)`
`" "stackrel("—————————————)`

Two As `2x-1=0` gives `x=1/2` put `1/2` at the left.

`" "1/2|color(white)(X)8" "color(white)(X)-20color(white)(XXX)2" "" "-2`
`color(white)(xxx)|" "color(white)(XX)`
`" "stackrel("———————————————)`

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

`1/2|color(white)(X)8" "color(white)(X)-20color(white)(XX)2" "" "-2`
`color(white)(x)|" "color(white)(X)`
`" "stackrel("—————————————)`
`color(white)(x)|color(white)(X)color(red)8`

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

`1/2|color(white)(X)8" "color(white)(X)-20color(white)(XxX)2" "" "-2`
`color(white)(x)|" "color(white)(xxxxxx)4`
`" "stackrel("——————————————)`
`color(white)(x)|color(white)(X)color(blue)8`

Five Add down the column.

`1/2|color(white)(X)8" "color(white)(X)-20color(white)(XxX)2" "" "-2`
`color(white)(x)|" "color(white)(xxxxxx)4`
`" "stackrel("——————————————)`
`color(white)(x)|color(white)(X)color(blue)8color(white)(XXx)color(red)-16`

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

`1/2|color(white)(X)8" "color(white)(X)-20color(white)(XxX)2" "" "-2`
`color(white)(x)|" "color(white)(xxxxxx)4color(white)(XX)-8color(white)(XX)-3`
`" "stackrel("——————————————)`
`color(white)(x)|color(white)(X)color(blue)8color(white)(XXX)color(red)(-16)color(white)(xX)color(red)(-6)color(white)(XXX)color(red)(-5)`

Hence, Quotient is `8x^2-16x-6` and remainder is `-5`.

in other words `8x^3-20x^2+2x-2=(x-1/2)(8x^2-16x-6)-5`

or `8x^3-20x^2+2x-2=(2x-1)(4x^2-8x-3)-5`