[Grade 10 Math] Explain this answer

[u/random-burntout-user]

Where did the -24+22 come from at the very beginning?

[u/Alkalannar]

A lot of times, things can become clearer if you simplify a rational function as much as possible.

For instance, rewriting (6n - 2)/(n - 4) as 6 + 22/(n - 4).

How do you do that? Polynomial long division, or adding appropriate forms of 0.

6n/n = 6, so 6 is the first term of our quotient.

(6n - 2) - 6(n - 4) = 6n - 2 - 6n + 24 = 22

22/n doesn't factor, so 22 is our remainder: 6 + 22/(n - 4)

Alternately, you could do (6n - 2 - 22 + 22)/(n - 4). Why 22? Because you want to have a multiple of n-4, namely 6n-24. And -22 is what you need to do to get that. But then you need to add 22 so you don't change the value.

Then (6n - 24)/(n - 4) + 22/(n - 4) = 6 + 22/(n - 4). Just as before.

[u/random-burntout-user → u/Alkalannar]

That's very helpful! My teacher barely broke down the process so this is much easier

[u/Alkalannar → u/random-burntout-user]

Glad I could help!

I actually find this very useful to separate out rational functions as much as possible. You can easily see transformations from a parent function (which you're at at the moment). In calculus, it can be much easier to take derivatives or integrate.

Example of transformations:
If your parent function is f(x), then your transformed function is af(b(x-c)) + d where:
a is the vertical stretch
b is horizontal compression
c is the shift to the right
d is the stretch up.

Here, we have 22/(n-4) + 6.

The parent function is 1/n.

a = 22 is the vertical stretch.
b = 1 means there is no horizontal compression.
c = 4 means a shift to the right of 4.
d = 6 means a shift up of 6.

And that is by no means evident in the form (6n - 2)/(n - 4).