A product term that leaves out a variable covers several rows. Expanding it lists those rows as full minterms.
The algebra. Multiply the term by for each missing variable X. That factor equals 1, so nothing changes, but every term now contains every variable. For a sum term, the dual trick adds (which is 0) and splits it with OR over AND: = .
The shortcut. Write each term in cube notation, with a dash for each missing variable. Each dash can be 0 or 1, so a term with k dashes covers 2ᵏ rows.
- For a product, the pattern shows where it is 1 (1 → plain, 0 → barred).
- For a sum, the pattern shows where it is 0 (0 → plain, 1 → barred).
Then take the union of all the rows and list each once. Terms often overlap, so never just add up 2ᵏ for each term.
Expansion is how you get from any expression to a canonical form, which you can then convert, complement or simplify systematically.