الأسبوع 6 · الحالات الخاصة: المربع الكامل والفرق بين مربعين ومجموع مكعبين والفرق بين مكعبين · 3 دروس
3
Sum and difference of two cubes
Mathematics Prep 2 Term 1, Unit 2, Lesson 3
GoalFactorise a³ + b³ and a³ − b³, watch the signs in the second bracket, and use them in volume problems.
New words
sum of two cubes · a³ + b³ = (a + b)(a² − ab + b²)difference of two cubes · a³ − b³ = (a − b)(a² + ab + b²)perfect cube · a number or term that is something cubed: 27x³ = (3x)³opposite sign · the middle sign of the long bracket is opposite to the sign in the short one
Explanation
Expand (a − b)(a² + ab + b²): a³ + a²b + ab² − a²b − ab² − b³ = a³ − b³. So a³ − b³ = (a − b)(a² + ab + b²), and in the same way a³ + b³ = (a + b)(a² − ab + b²). Notice the pattern: the short bracket has the same sign as the original, the middle term of the long bracket has the opposite sign, and the last term is always plus. For 27x³ − 1: a = 3x and b = 1, so 27x³ − 1 = (3x − 1)(9x² + 3x + 1). For x³ + 8: (x + 2)(x² − 2x + 4). The long bracket a² ∓ ab + b² cannot be factorised further over the real numbers. Careful with the signs: x³ − 8 = (x − 2)(x² + 2x + 4), not (x − 2)(x² − 2x + 4).
Expression
Factorised
a³ + b³
(a + b)(a² − ab + b²)
a³ − b³
(a − b)(a² + ab + b²)
27x³ − 1
(3x − 1)(9x² + 3x + 1)
x³ + 8
(x + 2)(x² − 2x + 4)
Worked examples
Factorise 8x³ + 125. 8x³ = (2x)³ and 125 = 5³, so a = 2x and b = 5: (2x + 5)(4x² − 10x + 25). The middle term is −ab = −2x × 5 = −10x.
Factorise 2x³ − 16. Take out 2: 2(x³ − 8) = 2(x − 2)(x² + 2x + 4).
A solid cube of edge 5 cm has a cube of edge 2 cm cut out of a corner. The volume left is 5³ − 2³ = 125 − 8 = 117 cm³. By the formula: (5 − 2)(25 + 10 + 4) = 3 × 39 = 117. Both ways agree.
Class activity
Cube check. Pick a = 4 and b = 1. Work out a³ − b³ and (a − b)(a² + ab + b²). Then do a³ + b³ and (a + b)(a² − ab + b²). Do the answers agree? Try a = 3 and b = 2 too.
