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Just to be upfront, this is a homework question, I already know the answer, but I can't figure out how to get there or the logic behind the hint, which is really what I'm after. Please don't solve it for me, just give me some pointers in the right direction or links to better instructions.

The problem:

Factor the expression $x^3 - 3x^2 + 4$

The hint the book provides "subtract and add 1, then factor by grouping"

The given answer is $(x+1)(x-2)^2$

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$-3x^2=x^2-4x^2$ – pedja Oct 25 '11 at 17:35
@pedja is that what they mean by subtract and add 1? – Marshall Brekka Oct 25 '11 at 17:41
no,they mean: $(x^3+1)+(-3x^2+4-1)$ – pedja Oct 25 '11 at 17:48
If you don't see it otherwise, you can try the rational root theorem, saying that any rational roots for this polynomial are among $\pm 1, \pm 2, \pm 4$. Once you find one that works, divide it out and you have a quadratic. – Ross Millikan Oct 25 '11 at 17:49
@Marshall, pedja yes because you are just writing 3 as (4-1) in front of $x^2$, precisely 3=$(3+1)-1$ :) – Tapu Oct 25 '11 at 17:50
up vote 2 down vote accepted


$x^3 - 3x^2 + 4 = x^3 + 1 -3x^2 + 3 = (x + 1)(...) - 3(x^2 - 1) = ...$

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Then you'll use the "sum of cubes" and "difference of squares" formula. – The Chaz 2.0 Oct 25 '11 at 17:54
wow thank you so much! Also it was really helpful that you suggested the 2 formulas, I just watched some videos on both of those and it really helped. – Marshall Brekka Oct 25 '11 at 18:15
pedja's shortcut is definitely helpful, but many hints like that require you to already know how to do the problem! Hopefully in the future you'll be able to "spot" such shortcuts. – The Chaz 2.0 Oct 25 '11 at 18:33

Hint: use @pedja,s hint (this is the best, I can think of).

Else, split $4$ as $4=3+1$ and go through a longer route.

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