# How do you expand (a-b)^3?

Feb 22, 2017

${a}^{3} - 3 {a}^{2} b + 3 a {b}^{2} - {b}^{3}$

#### Explanation:

Use the Binomial expansion (note the exponents sum to the power in each term):
${\left(x + y\right)}^{3} {=}_{3} {C}_{0} {x}^{3} {y}^{0} {+}_{3} {C}_{1} {x}^{2} {y}^{1} {+}_{3} {C}_{2} {x}^{1} {y}^{2} {+}_{3} {C}_{3} {x}^{0} {y}^{3}$

Remember 3! = 3*2*1 = 6, 2! = 2*1 = 2, 1! = 1 and 0! = 1

_3C_0 = (3!)/((3-0)!(0!)) = (3!)/((3)!1) = 1

_3C_1 = (3!)/((3-1)!(1!)) = (3!)/((2)!1) = (3*2!)/(2!) = 3

_3C_2 = (3!)/((3-2)!(2!)) = (3!)/((1)!2!) = (3*2!)/(2!) = 3

_3C_3 = (3!)/((3-3)!(3!)) = (3!)/(0!*3!) = 1

Note: ${\left(a - b\right)}^{3} = {\left(a + \left(- b\right)\right)}^{3}$

Substitute into the Binomial expansion formula,
let $x = a$ and $y = - b$:

${\left(a - b\right)}^{3} = {a}^{3} + 3 {a}^{2} {\left(- b\right)}^{1} + 3 a {\left(- b\right)}^{2} + {\left(- b\right)}^{3}$

$= {a}^{3} - 3 {a}^{2} b + 3 a {b}^{2} - {b}^{3}$