Question

How do you divide `(x^4-81)/(x-3)` using synthetic division?

Answer 1

`(x^4 -81) div (x-3) = x^3 +3x^2 +9x +27`

Explanation:

In synthetic division we only use the numerical coefficients.

Make sure that they are in descending order of the powers of x.
In this case we have `" "x^4" "x^3" "x^2" "x^1" " x^0`

Write a `0` for any missing term.

If `x-3 = 0 rarr x = 3`

`color(red)(3)` is used at the side.
`color(white)(xxxxxxxxxxxxx)" "(x^4" "x^3" "x^2" "x^1" " x^0)`

Coefficients :`rarrcolor(white)(xx)" "1" "0" "0" "0" "-81`

`color(white)(xxxxxxxxxxxx)color(red)(3)color(white)(xx)ul(darr " "color(blue)(3)" "color(purple)(9)" "color(mediumpurple)(27) " "color(darkcyan)(81)`
`color(white)(xxxxxxxxxxxxx)color(white)(xxx)1" "color(lime)(3)" "color(orange)(9)" "color(magenta)(27)" "0 larr` remainder

`color(white)(xxxxxxxxxxxxx)" "x^3" "x^2" "x^1" " x^0`

Steps: Follow the colours...
bring down the 1.
`color(red)(3)xx1 = color(blue)(3)" add " 0+color(blue)(3) = color(lime) (3)`
`color(red)(3)xxcolor(lime)(3) = color(purple)(9)" add " 0+color(purple)(9) =color(orange)(9)`
`color(red)(3)xxcolor(orange)(9) =color(mediumpurple)(27)" add "0 + color(mediumpurple)(27) = color(magenta)(27)`
`color(red)(3)xxcolor(magenta)(27) = color(darkcyan)(81)" add "-81+ color(darkcyan)(81) = 0 larr` the remainder

The numbers in the bottom row are now the numerical coefficients of the quotient.

`x^4 divx = x^3` So the first term will be `x^3`

The quotient is:

`1x^3 + color(lime)(3)x^2 + color(orange)(9)x + color(magenta)(27) " rem " 0`