Without an exhaustive search, let's try a synthetic division approach. Start by dividing the polynomial by x−1. If we get a remainder of zero, then x−1 is a factor.
After performing synthetic division with x−1, it appears x−1 is not a factor. Let's try x+1:
Using synthetic division for x+1, we find that x+1 is also not a factor.
To speed up the process, using a calculator or software to determine roots will be beneficial. However, if done manually, we'd continue this process trying other potential factors.
If a factor is found (say x−r), we can then further factor the quotient polynomial obtained after division.
However, without computational tools, finding a factor for higher-degree polynomials can be tedious. If a particular value of
x is not immediately apparent as making f(x)=0, then it's often beneficial to turn to software or calculators for help.