Question

How do you divide `(-x^3+13x^2-x-5)/(x-4)`?

Answer 1

`-x^2+9x+35+135/(x-4)`

Explanation:

One way is to use the divisor as a factor in the numerator.

`color(magenta)"add / subtract "" the terms that occur as a consequence"`

`"consider the numerator"`

`color(red)(-x^2)(x-4)color(magenta)(-4x^2)+13x^2-x-5`

`=color(red)(-x^2)(x-4)color(red)(+9x)(x-4)color(magenta)(+36x)-x-5`

`=color(red)(-x^2)(x-4)color(red)(+9x)(x-4)color(red)(+35)(x-4)color(magenta)(+140)-5`

`rArr"quotient "=color(red)(-x^2+9x+35)" remainder "=135`

`rArr(-x^3+13x^2-x-5)/(x-4)=-x^2+9x+35+135/(x-4)`

Answer 2

`(-x^2+9x+35)+135`

Explanation:

For this question, use synthetic division (if you don't know what that is, check this out: http://www.purplemath.com/modules/synthdiv.htm)

x-4 is linear, so we are allowed to use synthetic division.

Your synthetic division should look something like this:

4| -1 13 -1 -5

     -4   36  140

-1   9   35   135

So we get that `(-x^3+13x^2-x-5)/(x-4) = (-x^2+9x+35)+135`

Alternatively, you can use the long division to do this, though it will take a bit longer (https://www.mathsisfun.com/algebra/polynomials-division-long.html)

Hope that helps!