# How do you find the product (2a-b)^3?

Jul 15, 2017

The solution is: $8 {a}^{3} - 12 {a}^{2} b + 6 a {b}^{2} - {b}^{3}$

#### Explanation:

There may be a more efficient and compact way, and someone may explain it, but I'd tend to just brute-force it. ;-)

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

Ignore the third bracket for now and do 'FOIL (first, outers, inners, lasts) on the first two brackets:

$\left(2 a - b\right) \left(2 a - b\right) \left(2 a - b\right) = \left(4 {a}^{2} - 2 a b - 2 a b + {b}^{2}\right) \left(2 a - b\right)$

Collect like terms:

$\left(4 {a}^{2} - 4 a b + {b}^{2}\right) \left(2 a - b\right)$

Now multiply each term in the left bracket by each term in the right:

$8 {a}^{3} - 4 {a}^{2} b - 8 {a}^{2} b + 4 a {b}^{2} + 2 a {b}^{2} - {b}^{3}$

Collect like terms again:

$8 {a}^{3} - 12 {a}^{2} b + 6 a {b}^{2} - {b}^{3}$

And we're done!