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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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  • $\begingroup$ $-3x^2=x^2-4x^2$ $\endgroup$
    – Peđa
    Oct 25, 2011 at 17:35
  • $\begingroup$ @pedja is that what they mean by subtract and add 1? $\endgroup$ Oct 25, 2011 at 17:41
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    $\begingroup$ no,they mean: $(x^3+1)+(-3x^2+4-1)$ $\endgroup$
    – Peđa
    Oct 25, 2011 at 17:48
  • $\begingroup$ 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. $\endgroup$ Oct 25, 2011 at 17:49
  • $\begingroup$ @Marshall, pedja yes because you are just writing 3 as (4-1) in front of $x^2$, precisely 3=$(3+1)-1$ :) $\endgroup$
    – Tapu
    Oct 25, 2011 at 17:50

2 Answers 2

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HINT

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

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    $\begingroup$ Then you'll use the "sum of cubes" and "difference of squares" formula. $\endgroup$ Oct 25, 2011 at 17:54
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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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