# Boole's Expansion Theorem > [!Example] Boole's Expansion Theorem > *Boole's expansion theorem*, also known as *Shannon expansion* or *decomposition*, states that for any [[Boolean function]] $F$, > $F(X_{1},X_{2},\dots,X_{n})=X_{1}F(1,X_{2},\dots,X_{n})+\bar{X_1}F(0,X_{2},\dots, X_{n})$ > The [[Duality#Boolean Algebra|dual]] form of the theorem is > $F(X_{1},X_{2},\dots,X_{n})=(X_{1}+F(0,X_{2},\dots,X_{n}))(\bar{X_{1}}+F(1,X_{2},\dots,X_{n}))$ Repeatedly applying the theorem or its dual for each argument gives the [[Boolean Function#Sum-of-product or product-of-sums form|sum-of-products or product-of-sums form]] of the function, respectively.