Question

How do you long divide `(x^3-13x -12)/ (x-4)`?

Answer 1

The quotient is `=x^2+4x+3` and the remainder is `=0`

Explanation:

Let's perform the long division

`color(white)(aaaa)` `x-4` `|` `color(white)(aaaa)` `x^3+0x^2-13x-12` `color(white)(aaaa)` `|` `x^2+4x+3`

`color(white)(aaaaaaaaaaaaaa)` `x^3-4x^2`

`color(white)(aaaaaaaaaaaaaaa)` `0+4x^2-13x`

`color(white)(aaaaaaaaaaaaaaaaa)` `+4x^2-16x`

`color(white)(aaaaaaaaaaaaaaaaaaa)` `+0+3x-12`

`color(white)(aaaaaaaaaaaaaaaaaaaaaaa)` `+3x-12`

`color(white)(aaaaaaaaaaaaaaaaaaaaaaaaa)` `+0-0`

Therefore,

`(x^3-13x-12)/(x-4)=x^2+4x+3`

Answer 2

`x^2+4x+3`

Explanation:

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

`"consider the numerator"`

`color(red)(x^2)(x-4)color(magenta)(+4x^2)-13x-12`

`=color(red)(x^2)(x-4)color(red)(+4x)(x-4)color(magenta)(+16x)-13x-12`

`=color(red)(x^2)(x-4)color(red)(+4x)(x-4)color(red)(+3)(x-4)color(magenta)(+12)-12`

`=color(red)(x^2)(x-4)color(red)(+4x)(x-4)color(red)(+3)(x-4)`

`"quotient "=color(red)(x^2+4x+3)," remainder "=0`

`rArr(x^3-13x-12)/(x-4)=x^2+4x+3`