Question

How do you simplify `(4x + 5x ^ { 3} + 10x ^ { 2} + 8) \div ( 5x ^ { 2} - 1)`?

Answer 1

`x+2 + (5x+10)/(5x^2-1)`

Explanation:

After rearranging the dividend into standard form, you can perform long polynomial division:
`color(white)("xxxxxxxx")ul(color(white)("xx")xcolor(white)("xx")+2color(white)("xxxxxxxxxxxxxx"))`
`5x^2-1color(white)("x"))color(white)("x")5x^3color(white)("x")+10x^2color(white)("x")+4xcolor(white)("x")+8`
`color(white)("xxxxxxxxx")ul(5x^3color(white)("xxxxxxxx")-1xcolor(white)("xxxxx"))`
`color(white)("xxxxxxxxxxxxxxx")10x^2color(white)("x")+5xcolor(white)("x")+8`
`color(white)("xxxxxxxxxxxxxxx")ul(10x^2color(white)("xxxxxxx")-2)`
`color(white)("xxxxxxxxxxxxxxxxxxxxxx")5xcolor(white)("x")+10`

As far as I can see the dividend and divisor have no simple common factors, so this is probably the best that can be done (although it could be argued that this is not really a simplification)