Question

How do you divide `(x^2 - 2x - 15)/(x + 3)` using polynomial long division?

Answer 1

Answer 2 of 2

`x-5`

Have a look at the method. It shows a useful 'trick'.

Explanation:

Given: `(x^2-2x-15)/(x+3)........................(1)`

Not all questions permit this approach of solution!

Consider `color(white)(..)x^2-2x-15`

This can be factored into:

`(x-5)(x+3).............................(2)`

Substitute expression (2) into expression (1)

`((x-5)(x+3))/(x+3)`

Write as: `(x+3)/(x+3) xx (x-5)`

But `(x+3)/(x+3)` has the value of 1 giving

`1xx (x-5)`

`x-5`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Foot note")`

Consider: `(x+3)/(x+3)`

If you were investigating values then this produces a problem.
You are not mathematically allowed to divide by 0.

So `(x+3)/(x+3)` is 'Undefined' at `x=-3`

For this very reason `0/0` does `underline(color(red)("not equal 1"))`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Answer 2

Answer 1 of 2

Using polynomial long division.

`x-5`

Explanation:

Given: `(x^2-2x-15) -:(x+3)`

`color(white)("ddddddd.dd") x^2-2x-15`
`color(magenta)(x)(x+3)-> ul(x^2+3x larr" Subtract")`
`color(white)("dddddddddd") 0 color(white)(",") -5x-15`
`color(magenta)(-5)(x+3)-> color(white)("d") ul(-5x-15 larr" Subtract")`
`color(white)("dddddddddddddd") 0+0`

`(x^2-2x-15) -:(x+3) = color(magenta)(x-5)`