Complex Conjugate Factors
Summary: Euler’s argument (§30–§31) that complex linear factors of a real polynomial come in pairs whose product is a real quadratic factor.
Sources: chapter2
Last updated: 2026-04-23
Even count of complex factors
“In every equation the number of complex roots is even, so that the function has either no complex factor, or it has two or four or six, etc.” (source: chapter2, §30). If is the product of the real linear factors of , then must be real, so the remaining complex factors must combine into a real product.
Two complex factors → one real quadratic
If has exactly two complex linear factors, their product is a real quadratic, since removing the real factors from a real polynomial leaves a real quotient (source: chapter2, §30).
Four complex factors → two real quadratics (§31)
Euler considers a real polynomial that does not split into two real quadratic factors. He writes it as the product of two complex quadratics in conjugate form:
Expanding and solving gives four complex linear factors. Pairing the first with the third (and the second with the fourth), and letting and , each paired product turns out to be
which Euler observes is real. Thus , assumed not to split into two real quadratic factors, in fact does. By contradiction, every real quartic splits into two real quadratic factors.
Beyond degree four
Euler admits the same explicit construction does not go through in higher degree:
Although the same method of proof is not valid for higher powers, nevertheless, there is no doubt that the same property holds for any number of complex factors. (source: chapter2, §32)
The general case is taken as a working hypothesis and used to justify the §32 decomposition of any real polynomial into real linear and quadratic factors. See fundamental-theorem-of-algebra.