# What is the rule for cubing a binomial?

Oct 26, 2015

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

#### Explanation:

The coefficients $1 , 3 , 3 , 1$ can be found as a row of Pascal's triangle:

For other powers of a binomial use a different row of Pascal's triangle.

For example:

${\left(a + b\right)}^{5} = {a}^{5} + 5 {a}^{4} b + 10 {a}^{3} {b}^{2} + 10 {a}^{2} {b}^{3} + 5 a {b}^{4} + {b}^{5}$

How about ${\left(2 x - 5\right)}^{3}$ or similar?

Let $a = 2 x$ and $b = - 5$ to find:

${\left(2 x - 5\right)}^{3}$

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

$= {\left(2 x\right)}^{3} + 3 {\left(2 x\right)}^{2} \left(- 5\right) + 3 \left(2 x\right) {\left(- 5\right)}^{2} + {\left(- 5\right)}^{3}$

$= 8 {x}^{3} - 60 {x}^{2} + 150 x - 125$