Factoring a polynomial with 4 terms where factoring by grouping does not work?

Question

Ok so I am instructed to factor $t^3 -10t^2 +27t -18$ However, factoring by grouping does not work in this case? I know I can guess and check via the rational zero theorem. BUT THAT SEEMS SO TEDIOUS. Are there any other methods possible?

Answer 1

While in this case you'll hit the jackpot if you try $t=1$, the rational root theorem works well even if the smallest root would have large prime factors and would not come up a long time after the end of the universe when trying out the candidates one by one. To speed up the process, you should consider a change of variable after each negative result. E.g. substituting $t=-1$ yields -56. This means that substituting $t = u - 1$ and rewriting the polynomial in powers of $u$ will yield a constant term of $-56$. The roots in terms of $u$ must thus be divisors of $56 = 7*8$, In terms of $t$ these possible roots are thus of the form $\pm 2^n 7^m - 1$ with $0\leq n\leq 3$ and $0\leq m\leq 1$. But any root must also be a divisor of 18, this leaves you with the candidates:

$$t = -9,-3,-2,1,3,6$$

which is a considerably shorter list than the original list of all the divisors of 18.