Recall: the square formulae
- (a + b)² = a² + 2ab + b² — e.g. (x + 4)² = x² + 8x + 16
- (a − b)² = a² − 2ab + b² — e.g. (y − 5)² = y² − 10y + 25
- (a + b)(a − b) = a² − b² — e.g. (m + 6)(m − 6) = m² − 36
Quick square: 103² = (100 + 3)² = 10000 + 600 + 9 = 10609.
Recall: the square formulae
In the app you can move this figure with your finger.
Open in the appProduct (x + a)(x + b)
A square garden of side x m is made 5 m longer and 3 m wider. The new garden is (x + 5) × (x + 3). Its area is 4 pieces: the old square, two strips and a small corner rectangle.
When two binomials start with the same x, their product can be written straight from one formula.
In the figure: (x + 5)(x + 3): four parts x² + 5x + 3x + 15 = x² + 8x + 15 (the sum of 5 and 3 is 8, the product 15).
Formula: (x + a)(x + b) = x² + (a + b)x + ab
In words: x² , then the sum of the two numbers × x , then the product of the two numbers.
Product (x + a)(x + b)
In the app you can move this figure with your finger.
Open in the app- Easy: (x + 5)(x + 3) = x² + (5 + 3)x + 5 × 3 = x² + 8x + 15
- Medium: (x − 4)(x + 7) = x² + (−4 + 7)x + (−4) × 7 = x² + 3x − 28
- Exam-type: (y − 6)(y − 2) = y² + (−6 − 2)y + (−6) × (−2) = y² − 8y + 12
Quick numbers: 52 × 48 = (50 + 2)(50 − 2) = 2500 − 4 = 2496.
Take the + / − sign along with each number.
In (x − 4)(x + 7) the sum is −4 + 7 = 3, not 4 + 7.
Use this formula only when both binomials start with the same term x; first write the sum and product of a and b. Memory clue: (x + a)(x + b) = x² + (sum)x + product: "sum in the middle, product at the end".
Spot the difference
| Point | (x + 3)(x + 5) | (x + 3)(x − 3) |
|---|---|---|
| Sum of numbers | 8 | 0 |
| Product of numbers | 15 | −9 |
| Answer | x² + 8x + 15 | x² − 9 |
Practice questions
- 1(x + 2)(x + 9)
- 2(m − 3)(m + 8)
- 3(y − 5)(y − 4)
- 4(p + 11)(p − 6)
- 5(a − 7)(a − 7)
- 6Find 61 × 59 using the formula.
- 7Simplify (x + 3)(x + 5) − (x + 1)(x + 7).
- 8If (x + 5)(x + a) = x² + 9x + 5a, find a.
Show answers (try them yourself first)
- 1(x + 2)(x + 9) = x² + 11x + 18
- 2(m − 3)(m + 8) = m² + 5m − 24
- 3(y − 5)(y − 4) = y² − 9y + 20
- 4(p + 11)(p − 6) = p² + 5p − 66
- 5(a − 7)(a − 7) = a² − 14a + 49
- 661 × 59 = (60 + 1)(60 − 1) = 3600 − 1 = 3599
- 7(x + 3)(x + 5) − (x + 1)(x + 7) = (x² + 8x + 15) − (x² + 8x + 7) = 8
- 85 + a = 9, so a = 4.
Cube of (a + b)
A cube of side 2 cm gets a 1 cm layer all round and becomes a 3 cm cube: 27 = 8 + 12 + 6 + 1 — old cube, three slabs, three edge bars and one corner.
Multiplying (a + b) three times gives its cube. It has 4 terms.
In the figure: A 2 cm cube plus a 1 cm layer = a 3 cm cube: 8 (blue) + 12 + 6 + 1 = 27; (a + b)³ = a³ + 3a²b + 3ab² + b³ (here a = 2, b = 1).
Formula: (a + b)³ = a³ + 3a²b + 3ab² + b³
In words: cube of first + 3 × (first² × second) + 3 × (first × second²) + cube of second.
Cube of (a + b)
In the app you can move this figure with your finger.
Open in the app- Easy: (x + 2)³ = x³ + 3 × x² × 2 + 3 × x × 2² + 2³ = x³ + 6x² + 12x + 8
- Medium: (2m + 3)³ = (2m)³ + 3 × (2m)² × 3 + 3 × 2m × 3² + 3³ = 8m³ + 36m² + 54m + 27
- Exam-type: (3a + 2b)³ = (3a)³ + 3 × (3a)² × 2b + 3 × 3a × (2b)² + (2b)³ = 27a³ + 54a²b + 36ab² + 8b³
Quick numbers: 11³ = (10 + 1)³ = 1000 + 300 + 30 + 1 = 1331.
All 4 terms of (a + b)³ are positive; the signs do not alternate.
(2m)³ = 8m³, not 2m³. Cube the 2 as well.
The two middle terms have the numbers 3 and 3. Memory clue: Coefficients of the cube: 1, 3, 3, 1; the middle two terms carry 3.
Spot the difference
| Point | (a + b)³ | a³ + b³ |
|---|---|---|
| Number of terms | 4 | 2 |
| Middle terms | 3a²b + 3ab² | none |
| Example with x and 1 | x³ + 3x² + 3x + 1 | x³ + 1 |
| Are they equal | no | no — (a + b)³ ≠ a³ + b³ |
Practice questions
- 1(y + 1)³
- 2(x + 3)³
- 3Find 12³ using the formula.
- 4(3p + 1)³
- 5(2a + 5)³
- 6(m + 2n)³
- 7Simplify (x + 1)³ + (x − 1)³.
- 8If a + b = 5 and ab = 6, find a³ + b³.
Show answers (try them yourself first)
- 1(y + 1)³ = y³ + 3y² + 3y + 1
- 2(x + 3)³ = x³ + 9x² + 27x + 27
- 312³ = (10 + 2)³ = 1000 + 600 + 120 + 8 = 1728
- 4(3p + 1)³ = 27p³ + 27p² + 9p + 1
- 5(2a + 5)³ = 8a³ + 60a² + 150a + 125
- 6(m + 2n)³ = m³ + 6m²n + 12mn² + 8n³
- 7(x + 1)³ + (x − 1)³ = (x³ + 3x² + 3x + 1) + (x³ − 3x² + 3x − 1) = 2x³ + 6x
- 8a³ + b³ = (a + b)³ − 3ab(a + b) = 125 − 90 = 35. (a = 2, b = 3: 8 + 27 = 35)
Cube of (a − b)
Take a 1 cm layer off a 3 cm cube and a 2 cm cube remains: 27 − 19 = 8.
In (a − b)³ the signs alternate: + − + −.
In the figure: A 3 cm cube − a 1 cm layer (red) = a 2 cm cube: 27 − 19 = 8; (a − b)³ = a³ − 3a²b + 3ab² − b³ (here a = 3, b = 1).
Formula: (a − b)³ = a³ − 3a²b + 3ab² − b³
In words: Same as (a + b)³, but minus wherever the power of b is odd.
Cube of (a − b)
In the app you can move this figure with your finger.
Open in the app- Easy: (x − 2)³ = x³ − 3 × x² × 2 + 3 × x × 2² − 2³ = x³ − 6x² + 12x − 8
- Medium: (3y − 1)³ = (3y)³ − 3 × (3y)² × 1 + 3 × 3y × 1² − 1³ = 27y³ − 27y² + 9y − 1
- Exam-type: (5m − 2n)³ = (5m)³ − 3 × (5m)² × 2n + 3 × 5m × (2n)² − (2n)³ = 125m³ − 150m²n + 60mn² − 8n³
Quick numbers: 19³ = (20 − 1)³ = 8000 − 1200 + 60 − 1 = 6859.
(a − b)³ is (a + b)³ with b replaced by −b.
The last term is −b³ (minus), not +b³.
Write (a + b)³ and put a minus sign before the terms with an odd power of b. Memory clue: In the cube of a difference the signs alternate: +, −, +, −.
Spot the difference
| Point | (a + b)³ | (a − b)³ |
|---|---|---|
| Signs | + + + + | + − + − |
| e.g. (x ± 1)³ | x³ + 3x² + 3x + 1 | x³ − 3x² + 3x − 1 |
Practice questions
- 1(m − 1)³
- 2(y − 4)³
- 3Find 99³ using the formula.
- 4(2x − 3)³
- 5(4a − 1)³
- 6(p − 3q)³
- 7Simplify (x + 2)³ − (x − 2)³.
- 8If a − b = 3 and ab = 10, find a³ − b³.
Show answers (try them yourself first)
- 1(m − 1)³ = m³ − 3m² + 3m − 1
- 2(y − 4)³ = y³ − 12y² + 48y − 64
- 399³ = (100 − 1)³ = 1000000 − 30000 + 300 − 1 = 970299
- 4(2x − 3)³ = 8x³ − 36x² + 54x − 27
- 5(4a − 1)³ = 64a³ − 48a² + 12a − 1
- 6(p − 3q)³ = p³ − 9p²q + 27pq² − 27q³
- 7(x + 2)³ − (x − 2)³ = (x³ + 6x² + 12x + 8) − (x³ − 6x² + 12x − 8) = 12x² + 16
- 8a³ − b³ = (a − b)³ + 3ab(a − b) = 27 + 90 = 117. (a = 5, b = 2: 125 − 8 = 117)
Square of (a + b + c)
Three strips of 3 cm, 4 cm and 5 cm are laid end to end: 12 cm. A square on that length has area 12² = 144.
The square of a three-term sum has 6 terms: three squares and three double products.
In the figure: A square of side a + b + c: three squares a², b², c² (blue) and six rectangles ab, bc, ca (two of each).
Formula: (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
In words: square of each + 2 × product of every pair.
Square of (a + b + c)
In the app you can move this figure with your finger.
Open in the app- Easy: (x + y + 2)² = x² + y² + 2² + 2 × x × y + 2 × y × 2 + 2 × 2 × x = x² + y² + 4 + 2xy + 4y + 4x
- Medium: (2m + n + 1)² = (2m)² + n² + 1² + 2 × 2m × n + 2 × n × 1 + 2 × 1 × 2m = 4m² + n² + 1 + 4mn + 2n + 4m
- Exam-type: (a − 2b + 3c)² = a² + (2b)² + (3c)² − 2 × a × 2b − 2 × 2b × 3c + 2 × 3c × a = a² + 4b² + 9c² − 4ab − 12bc + 6ac
Quick numbers: 104² = (100 + 3 + 1)² = 10000 + 9 + 1 + 600 + 6 + 200 = 10816.
(a + b + c)² has 6 terms: three squares and three double products.
If a term is negative its double products turn negative too (2 × a × (−2b)).
Pairs: (a,b) (b,c) (c,a) — exactly three. Do not miss one. Memory clue: Three squares + 2 × the products of the three pairs: "3 squares, 3 pairs".
Spot the difference
| Point | (a + b)² | (a + b + c)² |
|---|---|---|
| Number of terms | 3 | 6 |
| Squares | a², b² | a², b², c² |
| Double products | 2ab | 2ab, 2bc, 2ca |
| Example (x, 1) / (x, y, 1) | x² + 2x + 1 | x² + y² + 1 + 2xy + 2y + 2x |
Practice questions
- 1(x + y + 1)²
- 2Check (2 + 3 + 4)² with the formula and with 9².
- 3(a + b + 2)²
- 4(3p + q + 2)²
- 5(x − 2y + 3z)²
- 6Find 105² using (100 + 5)².
- 7If a + b + c = 10 and ab + bc + ca = 31, find a² + b² + c².
- 8Simplify (x + y + 1)² + (x − y − 1)².
Show answers (try them yourself first)
- 1(x + y + 1)² = x² + y² + 1 + 2xy + 2y + 2x
- 2(2 + 3 + 4)² = 4 + 9 + 16 + 12 + 24 + 16 = 81; 9² = 81 ✔
- 3(a + b + 2)² = a² + b² + 4 + 2ab + 4b + 4a
- 4(3p + q + 2)² = 9p² + q² + 4 + 6pq + 4q + 12p
- 5(x − 2y + 3z)² = x² + 4y² + 9z² − 4xy − 12yz + 6xz
- 6105² = (100 + 5)² = 10000 + 1000 + 25 = 11025
- 710² = 100, so a² + b² + c² = 100 − 2 × 31 = 38.
- 8(x + y + 1)² + (x − y − 1)² = (x² + y² + 1 + 2xy + 2y + 2x) + (x² + y² + 1 − 2xy + 2y − 2x) = 2x² + 2y² + 2 + 4y