# How do I use Pascal's triangle to expand a binomial?

Oct 31, 2015

Rows of Pascal's triangle provide the coefficients to expand ${\left(a + b\right)}^{n}$ as follows...

#### Explanation:

To expand ${\left(a + b\right)}^{n}$ look at the row of Pascal's triangle that begins $1 , n$. This provides the coefficients.

For example, ${\left(a + b\right)}^{4} = {a}^{4} + 4 {a}^{3} b + 6 {a}^{2} {b}^{2} + 4 a {b}^{3} + {b}^{4}$ from the row $1 , 4 , 6 , 4 , 1$

How about ${\left(2 x - 5\right)}^{4}$ ?

Let $a = 2 x$ and $b = - 5$.

Then:

${\left(2 x - 5\right)}^{4} = {\left(a + b\right)}^{4} = {a}^{4} + 4 {a}^{3} b + 6 {a}^{2} {b}^{2} + 4 a {b}^{3} + {b}^{4}$

$= {\left(2 x\right)}^{4} + 4 {\left(2 x\right)}^{3} \left(- 5\right) + 6 {\left(2 x\right)}^{2} {\left(- 5\right)}^{2} + 4 \left(2 x\right) {\left(- 5\right)}^{3} + {\left(- 5\right)}^{4}$

$= 16 {x}^{4} + 4 \left(8 {x}^{3}\right) \left(- 5\right) + 6 \left(4 {x}^{2}\right) \left(25\right) + 4 \left(2 x\right) \left(- 125\right) + \left(625\right)$

$= 16 {x}^{4} - 160 {x}^{3} + 600 {x}^{2} - 1000 x + 625$