Question

How do you divide `(4x^5 +8x^4 -8x^3 +4x^2 +x-8) / (5x^2 -4x+9)`?

Answer 1

Since coefficient of highest power of `x`, i.e ., `x^2` term in the denominator is `!=1`, therefore we need to divide using the long division method.
Quotient `=4/5x^3+56/25x^2-156/125x+1396/625`
Remainder`=13229/625x-17564/625`

Explanation:

`color(white)(WWWWWWW)4/5x^3+56/25x^2-156/125x+1396/625`
`5x^2-4x+9)bar(4x^5+8x^4-8x^3+4x^2+x-8)(`
`color(white)(WWWWWW)4x^5-16/5x^4+36/5x^3`
`color(white)(WWWWW)ul(-color(white)(iiW)+color(white)(WiW)-color(white)(WWWWW))`
`color(white)(WWWWWWWWW)56/5x^4-76/5x^3+" "4x^2`
`color(white)(WWWWWWWWW)56/5x^4-224/25x^3+504/25x^2`
`color(white)(WWWWWWWW)ul(-color(white)(WiiW)+color(white)(WiW)-color(white)(WWW))`
`color(white)(WWWWWWWWWWW)-156/25x^3+404/25x^2+x`
`color(white)(WWWWWWWWWWW)-156/25x^3+624/125x^2-1404/125x`
`color(white)(WWWWWWIWW)ul(color(white)(WiiW)+color(white)(WWW)-color(white)(iWW)+)`
`color(white)(WWWWWWWWWWWWW)1396/125x^2+1529/125x-8`
`color(white)(WWWWWWWWWWWWW)1396/125x^2-5584/625x+12564/625`
`color(white)(WWWWWWWiWW)ul(color(white)(WiiW)-color(white)(WWW)+color(white)(WiWW)-)`
`color(white)(WWWWWWWWWWWWWWWWW)13229/625x-17564/625`