
40 The Fundamental Theorem of Algebra
because a
2
+b
2
≥a
2
, so both of the main square roots on the right-hand side are real.
In 1742 Euler asserted, without proof, that every real polynomial can be decom-
posed into linear or quadratic factors with real coefficients; Bernoulli now erred the
other way, citing
x
4
−4x
3
+ 2x
2
+ 4x + 4
with zeros 1 +
p
2 +
√
−3, 1 −
p
2 +
√
−3, 1 +
p
2 −
√
−3, and 1 −
p
2 −
√
−3.
Euler responded, in a letter to his friend Christian Goldbach, that the four factors
occur as two complex conjugate pairs, and that the product of such a pair of factors
is a real quadratic. He showed this to be the case for Bernoulli’s proposed counterex-
ample. Goldbach ...