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Let M n (F) be the algebra of n × n matrices over an arbitrary field F. We consider linear maps Φ : M n (F) → M r (F) preserving matrices annihilated by a fixed polynomial f (x) = (x − a 1) ⋯ (x − a m) with m ≥ 2 distinct zeroes a 1 , a 2 , ... , a m ∈ F ; namely, f (Φ (A)) = 0 whenever f (A) = 0. Suppose that f (0) = 0 , and the zero set Z (f) = { a 1 , ... , a m } is not an additive group. Then Φ assumes the form (†) A ↦ S ( A ⊗ D 1 A t ⊗ D 2 0 s ) S − 1 , for some invertible matrix S ∈ M r (F) , invertible diagonal matrices D 1 ∈ M p (F) and D 2 ∈ M q (F) , where s = r − n p − n q ≥ 0. The diagonal entries λ in D 1 and D 2 , as well as 0 in the zero matrix 0 s , are zero multipliers of f (x) in the sense that λ Z (f) ⊆ Z (f). In general, assume that Z (f) − a 1 is not an additive group. If Φ (I n) commutes with Φ (A) for all A ∈ M n (F) , or if f (x) has a unique zero multiplier λ = 1 , then Φ assumes the form (†). The above assertions follow from the special case when f (x) = x (x − 1) = x 2 − x , for which the problem reduces to the study of linear idempotent preservers. It is shown that a linear map Φ : M n (F) → M r (F) sending disjoint rank one idempotents to disjoint idempotents always assume the above form (†) with D 1 = I p and D 2 = I q , unless M n (F) = M 2 (Z 2). [ABSTRACT FROM AUTHOR]