Algebra
Expanding Brackets in Algebra
Expanding an expression means removing brackets by multiplying each term inside the bracket by the term outside. It is the reverse process of factorising.
Expanding brackets is simply the algebraic version of the distributive law, a rule so fundamental it is usually introduced with whole numbers long before algebra: 3 × (4 + 5) gives the same answer whether you add first (3 × 9 = 27) or distribute first (12 + 15 = 27). The FOIL acronym (First, Outside, Inside, Last) for double brackets was popularised in American algebra textbooks in the 20th century as a memory aid, though the underlying distributive process it describes is centuries older and applies to any number of bracketed terms, not just two.
Algebra Through Area Models
Expanding brackets is just finding the area of a rectangle whose sides are the two brackets. This is the exact same idea as multiplying two numbers as a rectangle's area — except now one or both sides have an unknown length x. Splitting the rectangle into smaller pieces (one for each term) and adding up their areas gives the expanded expression automatically, with no memorised acronym required.
x(x + 3): picture a rectangle with height x and width (x + 3). Split the width into an x-part and a 3-part, giving two smaller rectangles.
(x + 3)(x + 2): now both sides are two-part brackets, so the rectangle splits into a 2-by-2 grid of four smaller regions — exactly the four products FOIL asks you to find.
Adding the four regions gives x² + 3x + 2x + 6 = x² + 5x + 6 — exactly matching the FOIL result worked out below. The First, Outside, Inside, Last terms in FOIL are literally the four rectangle regions: First (x²) is top-left, Outside (2x, from the x on the left times the 2 on the bottom) is bottom-left here, Inside (3x, from the 3 on the top times the x on the right) is top-right, and Last (6) is bottom-right.
Single Bracket Expansion
To expand a(b + c), multiply a by every term inside the bracket: a times b + a times c. Every term inside gets multiplied — no skipping!
3 times x + 3 times 4 = 3x + 12.
-2 times 3a = -6a. -2 times -5 = +10. Answer: -6a + 10.
4x times 2x = 8x squared. 4x times -3y = -12xy. 4x times 1 = 4x. Answer: 8x squared - 12xy + 4x.
Double Bracket Expansion (FOIL)
To expand (a + b)(c + d), multiply each term in the first bracket by each term in the second bracket, then collect like terms. The FOIL method helps: First, Outside, Inside, Last.
First: x times x = x squared. Outside: x times 2 = 2x. Inside: 3 times x = 3x. Last: 3 times 2 = 6. Total: x squared + 2x + 3x + 6 = x squared + 5x + 6.
2x times x = 2x squared. 2x times 4 = 8x. -1 times x = -x. -1 times 4 = -4. Total: 2x squared + 8x - x - 4 = 2x squared + 7x - 4.
Special Products
| Pattern | Result | Example |
|---|---|---|
| (a + b) squared | a squared + 2ab + b squared | (x+3) squared = x squared + 6x + 9 |
| (a - b) squared | a squared - 2ab + b squared | (x-4) squared = x squared - 8x + 16 |
| (a + b)(a - b) | a squared - b squared | (x+5)(x-5) = x squared - 25 |
Common Mistakes
- Forgetting to multiply the term outside by every term inside: 3(x + 4) does not equal 3x + 4.
- Sign errors when a negative is outside: -2(x - 3) = -2x + 6, not -2x - 6.
- Skipping the like-terms step after FOIL and leaving four terms instead of three.
Key Takeaways
- Every term inside the bracket must be multiplied by the term outside.
- For double brackets, use FOIL: First, Outside, Inside, Last, then collect like terms.
- Learn the three special patterns — they appear constantly in later algebra.