Minimize term without Karnaugh map

Question

I have the following term, that should get minimized with Boolean algebra (no Karnaugh map!):

(a ∧ ¬b ∧ c) ∨ (a ∧ c ∧ d) ∨ (b ∧ d)

I already figured out, that the minimzed term is as follows (everything without the middle term) and it makes totally sense to me:

(a ∧ ¬b ∧ c) ∨ (b ∧ d)

However when I try to get same result with boolean algebra I am not able to reproduce it.

My first step is the following:

(a ∧ c) ∧ (¬b ∨ d) ∨ (b ∧ d)

When I apply distributivity-law I end up with the first (non-minimized) term.

Edit: Besides the question above, more general: Can every term that is minimizable by a Karnaugh map be minimized with boolean algebra?

Answer 1

\begin{align} a\overline{b}c+acd+bd & = a\overline{b}c + acd(b+\overline{b}) + bd \\ & = a\overline{b}c + acdb+acd\overline{b} + bd \\ & = (a\overline{b}c + a\overline{b}cd) + (bd + bdac) \\ & = a\overline{b}c + bd \\ \end{align}