Grade 11 Binomial Expressions: Short Notes

11 ශ්‍රේණිය · 6 පරිච්ඡේදය · ද්විපද ප්‍රකාශන

26%

of Grade 11 term tests include it (16 of 61)

120

marks in past papers (16 questions)

9 min

full lesson with quick checks

🎯 What you'll learn

Expand (a + b)³ and (a − b)³ and explain where the terms come from
Expand cubes like (2x − 3)³ correctly
Use binomial cubes to find values like 102³ and 99³
Recognise an expansion as a cube and evaluate it
Find a³ + b³ or a³ − b³ from a ± b and ab

The 10-second challenge

At the canteen your friend bets you a fish bun that you can't find 102³ without a calculator before he finishes counting to ten. With one identity from this chapter you can write 1 061 208 in a few seconds, and keep your bun money.

From square to cube

You already know: (a + b)² = a² + 2ab + b².A cube is three equal factors: (a + b)³ = (a + b)(a + b)(a + b).So multiply the square by one more (a + b):(a + b)³ = (a + b)(a² + 2ab + b²).An expression with two terms, like a + b or 2x − 3, is a binomial expression.

✏️ Building (a + b)³

Expand (a + b)(a² + 2ab + b²).

1a × (a² + 2ab + b²) = a³ + 2a²b + ab²
2b × (a² + 2ab + b²) = a²b + 2ab² + b³
3Add and collect like terms: a³ + 3a²b + 3ab² + b³.

✅ a³ + 3a²b + 3ab² + b³

🧮 Cube of a sum

Four terms. Powers of a go down 3, 2, 1, 0 while powers of b go up 0, 1, 2, 3. The coefficients are 1, 3, 3, 1.

(a + b)³ = a³ + 3a²b + 3ab² + b³

See it as a real cube

Cut a cube of edge a + b at a on every edge. You get 8 blocks: one a³, three a²b slabs, three ab² rods and one b³. Their volumes add to (a + b)³.

🧮 Cube of a difference

Same as the sum, but put −b in place of b. The signs alternate: +, −, +, −. Odd powers of −b are negative.

(a − b)³ = a³ − 3a²b + 3ab² − b³

⚠️ The cube is NOT a³ + b³

Examiners still see (a + b)² written as a² + b², and the same slip with cubes. Test it with numbers: (1 + 2)³ = 3³ = 27, but 1³ + 2³ = 9. The middle terms 3a²b and 3ab² are what make up the difference. Always write all four terms.

✏️ A first expansion

Expand (x + 2)³.

1a = x, b = 2.
2x³ + 3(x²)(2) + 3(x)(2²) + 2³
3= x³ + 6x² + 12x + 8

✅ x³ + 6x² + 12x + 8

⚠️ Cube the whole term

(2x)³ is 2x × 2x × 2x = 8x³, not 2x³. Likewise (2x)² = 4x². Put brackets round each term before you cube or square it, then simplify. A quick check: put x = 1. (2 − 3)³ = −1 and 8 − 36 + 54 − 27 = −1 ✓

🧮 Sum and difference of cubes

Rearranged cube identities. Use them when you are given a + b (or a − b) and ab, but not a and b themselves.

a³ + b³ = (a + b)³ − 3ab(a + b) ; a³ − b³ = (a − b)³ + 3ab(a − b)

📌 Remember

• (a + b)³ = a³ + 3a²b + 3ab² + b³• (a − b)³ = a³ − 3a²b + 3ab² − b³ (signs alternate)• Four terms, coefficients 1, 3, 3, 1; never just a³ + b³.• Bracket and cube the whole term: (2x)³ = 8x³.• Use cubes for 102³-type sums and to find a³ ± b³ from a ± b and ab.

🧾 Quick revision sheet

(a + b)³	a³ + 3a²b + 3ab² + b³
(a − b)³	a³ − 3a²b + 3ab² − b³
Coefficients in a binomial cube	1, 3, 3, 1
(2x)³	8x³
a³ + b³ in terms of a + b and ab	(a + b)³ − 3ab(a + b)
a³ − b³ in terms of a − b and ab	(a − b)³ + 3ab(a − b)
(x + 1)³	x³ + 3x² + 3x + 1