# Factor By Grouping 3rd Degree Polynomial

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 $(x+1)(x-2)^2$

• $-3x^2=x^2-4x^2$ – Peđa Terzić 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: $(x^3+1)+(-3x^2+4-1)$ – Peđa Terzić 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

$x^3 - 3x^2 + 4 = x^3 + 1 -3x^2 + 3 = (x + 1)(...) - 3(x^2 - 1) = ...$
Else, split $4$ as $4=3+1$ and go through a longer route.