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During lesson we were given a starter activity which was to try and factorise polynomials and see what happened.

The polynomials were $x^3-8$, $x^3-3x^2+ x -3$ and $x^4 - 16$.

I could not work out what happened to them, and it's bugging me. If anyone can explain then it'd be helpful! Thank you.

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Have you tried applying the rational root test? –  kahen Nov 13 '10 at 14:20
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Is this the right polynomial? –  AD. Nov 13 '10 at 14:39

3 Answers 3

It's not clear exactly what's your problem, but the last polynomial factorizes like this: $$x^4-16=(x^2)^2-4^2=(x^2+4)(x^2-4)=(x^2+4)(x+2)(x-2).$$ For the first one, it's easy to find the root $x=2$, which means that you can factor out $x-2$.

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And for the second polynomial, $x = 3$ is a root and then divide by $x =3$. –  user1119 Nov 13 '10 at 15:52

HINT $\rm\ \ x-a \ $ divides $\rm\ x^n - a^n\ $ handles the 1st and 3rd. The 2nd is $\rm\ x^2\ (x-3) + (x-3)\ =\ \ldots$

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a^3 - b^3 = (a - b)(a^2 + ab + b^2). So x^3 - 8 = (x - 2)(x^2 + 2x + 4).
For x^3 - 3x^2 + x - 3, factor by grouping the first two terms together and taking out the GCF and grouping the second two and doing the same. This should give you
x^2(x - 3) + 1(x - 3). This gives you a greatest common factor of (x - 3) that you can now factor out.
(x - 3)(x^2 + 1). And you are done.

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