Precalculus · Problem 30
Rational Functions — Stitz & Zeager
Mathematics — Precalculus · complex-zeros-polynomial-factoring
Problem
Find all of the zeros of , then completely factor it over the real numbers and completely factor it over the complex numbers.Intuitive explanation
The key is to notice that this cubic has a hidden structure—try grouping pairs of terms to factor out common expressions. Once you find one real zero this way, you can divide it out and work with the remaining quadratic. That quadratic might factor nicely over the reals, or it might have complex roots that you'll need to express using the quadratic formula. The final factorizations will show how many real versus complex zeros exist.
Intuitive explanation
The key is to notice that this cubic has a hidden structure—try grouping pairs of terms to factor out common expressions. Once you find one real zero this way, you can divide it out and work with the remaining quadratic. That quadratic might factor nicely over the reals, or it might have complex roots that you'll need to express using the quadratic formula. The final factorizations will show how many real versus complex zeros exist.