Question

How do you simplify `\frac { x ^ { 3} + 5x ^ { 2} - 5x + 8} { x - 3}`?

Answer 1

Use long division to factorise the numerator

Explanation:

Using long division, we divide the numerator by the denominator to give ourselves a quadratic and a remainder.

`(x^3+5x^2-5x)/(x-3) =(x^2+8x+19)+65/(x-3)`

Answer 2

`x^2+8x+19+65/(x-3)`

Explanation:

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

`"consider the numerator"`

`color(red)(x^2)(x-3)color(magenta)(+3x^2)+5x^2-5x+8`

`=color(red)(x^2)(x-3)color(red)(+8x)(x-3)color(magenta)(+24x)-5x+8`

`=color(red)(x^2)(x-3)color(red)(+8x)(x-3)color(red)(+19)(x-3)color(magenta)(+57)+8`

`=color(red)(x^2)(x-3)color(red)(+8x)(x-3)color(red)(+19)(x-3)+65`

`"quotient "=color(red)(x^2+8x+19)," remainder "=65`

`>rArr(x^3+5x^2-5x+8)/(x-3)`

`=x^2+8x+19+65/(x-3)`