Factorize $2a^3 - b^3 - c^3$

Question

I need to factorize the expression $2a^3 - b^3 - c^3$. I see that one zero is achieved when $a=b=c$, but I can't find the factor(s).

Answer 1

$$2X^3-b^3-c^3=0$$

So $X=\sqrt[3]{\frac {b^3+c^3} 2}$ is a root, now do long division and use the quadratic formula.

Answer 2

I assume you're looking for factors that are polynomials with integer coefficients.

Your polynomial is not factorizable in this sense. To see why, look at what happens when $b=0,c=1$. You get $2a^3-1$, which you cannot factor over the integers.

Answer 3

Writing $$ 2a^3-b^3-c^3=a^3-b^3+a^3-c^3=(a-b)(a^2+ab+b^2)+(a-c)(a^2+ac+c^2) $$ We can see that we have simple factorizations if $a=b$ or $a=c$ or $b=c$ and also for $b+a=c$. But in general we cannot find a factorization for all real values of $a,b,c$.