Question

How do you divide `(2x^4 -x^3 - 9x^2 - x + 5)/ (5x^2 - 2)` using polynomial long division?

Answer 1

This really is long division but its layout is different to the conventional approach.

`2/5x^2-1/5x-41/25+(-35x+44)/(25 (5x^2-2))`

Explanation:

Given: `(color(brown)(2x^4-x^3-9x^2-x+5))/(color(green)(5x^2-2))`

Using place keepers of no value. Example: `0x^3`

`color(white)("ddddddddddddddd")color(brown)(2x^4-color(white)("d")x^3color(white)("d")-9x^2-x+5)`
`color(magenta)(+2/5x^2)color(green)((5x^2-2)) ->ul(2x^4+0x^3-4/5x^2larr" Subtract")`
`color(white)("ddddddddddddddddd")0-x^3-41/5x^2-x+5`
`color(magenta)(-1/5)color(green)((5x^2-2)) ->color(white)("d.dd")ul( -x^3+color(white)("d.")0x^2+2/5xlarr" Subtract")`
`color(white)("ddddddddddddddddddddd")0-41/5x^2-7/5x+5`
`color(magenta)(-41/25)color(green)((5x^2-2)) ->color(white)("dddddddd")ul(-41/5x^2+0x+81/25larr" Sub.")`
`color(white)("d")color(magenta)("Remainder "->color(white)("ddddddddddddd")0-7/5x+44/25)`

`color(magenta)("Set remainder as: "(-35x+44)/25)` giving:

`color(magenta)(2/5x^2-1/5x-41/25+[(-35x+44)/25 color(green)(-:(5x^2-2))])`

`color(magenta)(2/5x^2-1/5x-41/25+[(-35x+44)/(25 color(green)((5x^2-2)))])`

Answer 2

The remainder is `=(-7/5x+43/25)` and the quotient is `=(2/5x^2-x/5-41/25)`

Explanation:

Perform the polynomial long division

`color(white)(aaaa)` `2x^4-x^3-9x^2-x+5` `color(white)(aaaa)` `|` `5x^2-2`

`color(white)(aaaaaaaaaaaaaaaaaaaaaaaa)` `color(white)(aaaa)` `|` `2/5x^2-x/5-41/25`

`color(white)(aaaa)` `2x^4-0x^3-4/5x^2`

`color(white)(aaaa)` `0x^4-1x^3-41/5x^2-x`

`color(white)(aaaaaaaa)` `-1x^3-00x^2+2/5x`

`color(white)(aaaaaaaa)` `-0x^3-41/5x^2-7/5x+5`

`color(white)(aaaaaaaaaaaaa)` `-41/5x^2-7/5x+82/25`

`color(white)(aaaaaaaaaaaaaaa)` `-0x^2-7/5x+43/25`

Therefore,

`(2x^4-x^3-9x^2-x+5)/(5x^2-2)=(2/5x^2-x/5-41/25)+(-7/5x+43/25)/(5x^2-2)`

The remainder is `=(-7/5x+43/25)` and the quotient is `=(2/5x^2-x/5-41/25)`