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I managed to confuse the heck out of myself, so I'll post the impossible conclusion I recently made.

[tex]\frac{4x^3-2x^2+8x+8}{2x+1}[/tex]

I don't know why not, but this is impossible and I know it, but it seems to make perfect sense. Keep an eye on the 1 in the denominator and the 8 in the numerator.

[tex]\frac{4x^3-2x^2+8x+8x}{2x+x}[/tex]

See, it "feels" like it makes perfect sense to "uncancel" an invisible x. However, something doesn't feel right. I then do the following (which then gives me an even more impossible conclusion.)

[tex]\frac{4x^2-2x^2+8x+8x}{2x+1}[/tex]

I cancel the x in the denomenator in trade for bringing the cubic up stairs down to a square.

[tex]\frac{2x^2 + 16x}{2x+1}[/tex]

Subtracting and adding terms in the numerator.

Now I take 2x and cancel from both 16x and [tex]2x^2[/tex], which yields:

[tex]{x^2 + 16}[/tex]

I'm getting a sick feeling in my stomach for not being able to understand what's going wrong here, I've never had trouble with algebra, especially not with something like this. -_-; I think it's high time for a refresher course. Can someone target what I've done wrong here, and maybe point me to a good source to brush up on the basics? Many thanks in advance.

*I just noticed, after posting, that I made the 2x from 2x+1 vanish all together, instead of leaving a 1, that conclusion should have been [tex]{x^2 + 8}[/tex]; I still don't think that adds up though.

[tex]\frac{4x^3-2x^2+8x+8}{2x+1}[/tex]

I don't know why not, but this is impossible and I know it, but it seems to make perfect sense. Keep an eye on the 1 in the denominator and the 8 in the numerator.

[tex]\frac{4x^3-2x^2+8x+8x}{2x+x}[/tex]

See, it "feels" like it makes perfect sense to "uncancel" an invisible x. However, something doesn't feel right. I then do the following (which then gives me an even more impossible conclusion.)

[tex]\frac{4x^2-2x^2+8x+8x}{2x+1}[/tex]

I cancel the x in the denomenator in trade for bringing the cubic up stairs down to a square.

[tex]\frac{2x^2 + 16x}{2x+1}[/tex]

Subtracting and adding terms in the numerator.

Now I take 2x and cancel from both 16x and [tex]2x^2[/tex], which yields:

[tex]{x^2 + 16}[/tex]

I'm getting a sick feeling in my stomach for not being able to understand what's going wrong here, I've never had trouble with algebra, especially not with something like this. -_-; I think it's high time for a refresher course. Can someone target what I've done wrong here, and maybe point me to a good source to brush up on the basics? Many thanks in advance.

*I just noticed, after posting, that I made the 2x from 2x+1 vanish all together, instead of leaving a 1, that conclusion should have been [tex]{x^2 + 8}[/tex]; I still don't think that adds up though.

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