We find a class of algebras
A
\mathcal {A}
satisfying the following property: for every nontrivial noncommutative polynomial
f
(
X
1
,
…
,
X
n
)
f(X_1,\ldots ,X_n)
, the linear span of all its values
f
(
a
1
,
…
,
a
n
)
f(a_1,\ldots ,a_n)
,
a
i
∈
A
a_i\in \mathcal {A}
, equals
A
\mathcal {A}
. This class includes the algebras of all bounded and all compact operators on an infinite dimensional Hilbert space.