Question

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

Answer 1

`x^2+1+(7-5x)/(x^2+2)`

Explanation:

`"one way is to use the divisor as a factor in the numerator"`

`"consider the numerator"`

`color(red)(x^2)(x^2+2)color(magenta)(-2x^2)+3x^2-5x+9`

`=color(red)(x^2)(x^2+2)color(red)(+1)(x^2+2)color(magenta)(-2)-5x+9`

`=color(red)(x^2)(x^2+2)color(red)(+1)(x^2+2)+7-5x`

`"quotient "=color(red)(x^2+1)," remainder "=7-5x`

`rArr(x^4+3x^2-5x+9)/(x^2+2)=x^2+1+(7-5x)/(x^2+2)`