The Core Rule
Think of expanding like a handshake at a party: every term in the left bracket must shake hands (multiply) with every term in the right bracket. Nobody gets left out.
The Rule
For (a + b)(x + y + z), distribute each term in the left bracket:
= a(x + y + z) + b(x + y + z)
= ax + ay + az + bx + by + bz
Why does this work?
Multiplication distributes over addition — it's just the distributive law scaled up. When the brackets get bigger, the same principle applies; you just end up with more terms.
The FOIL Shortcut
When both brackets each contain exactly two terms, FOIL is a memory trick for the order you multiply — though it's really just the same general rule applied to a 2×2 situation.
(2x + 3)(x − 5)
FIRST → 2x × x = 2x²
OUTSIDE → 2x × (−5) = −10x
INSIDE → 3 × x = +3x
LAST → 3 × (−5) = −15
Combine: 2x² − 10x + 3x − 15 = 2x² − 7x − 15
Common Mistake
FOIL only works when each bracket has exactly
two terms. For (2x + 3)(x² + x − 5) you must use full distribution — there's no FOIL shortcut here.
Practice Question 1
Expand and simplify (3x − 4)(2x + 7).
Squaring & Cubing Brackets
When you see (a + b)² or (a + b)³, the trick is simple — rewrite them as repeated multiplication, then expand two at a time.
The Approach
- (3x + 2)² → write as (3x + 2)(3x + 2), then FOIL
- (a + b)³ → write as (a + b)(a + b)(a + b), expand two first, then multiply the result by the third
- For higher powers, use the Binomial Expansion (covered in a later chapter)
Worked Example — Squaring a bracket
Expand (3x + 2)²
1
Rewrite: (3x + 2)(3x + 2)
2
Apply FOIL: 9x² + 6x + 6x + 4
Critical Error to Avoid
(a + b)² ≠ a² + b². This is one of the most frequently dropped marks in algebra. Always expand fully — the middle term 2ab is never zero unless a or b is zero.