# How do you use the Binomial Theorem to expand (2x−1)^4?

Apr 27, 2018

$16 {x}^{4} - 32 {x}^{3} + 24 {x}^{2} - 8 x + 1$

#### Explanation:

Given: ${\left(2 x - 1\right)}^{4}$

We can say that:
${\left(2 x\right)}^{4} {\left(- 1\right)}^{0} + {\left(2 x\right)}^{3} {\left(- 1\right)}^{1} + {\left(2 x\right)}^{2} {\left(- 1\right)}^{2} + {\left(2 x\right)}^{1} {\left(- 1\right)}^{3} + {\left(2 x\right)}^{0} {\left(- 1\right)}^{4} =$

$16 {x}^{4} - 8 {x}^{3} + 4 {x}^{2} - 2 x + 1$

However we are not quite done, we need the coefficients in front of each term, I would use combinations unless you really want to use Pascal's triangle, which you can since this a small exponent.

Using pascal's triangle:
For row 4: the coefficients are $1 , 4 , 6 , 4 , 1$

$1 \cdot 16 {x}^{4} + 4 \cdot - 8 {x}^{3} + 6 \cdot 4 {x}^{2} + 4 \cdot - 2 x + 1 \cdot 1$

$16 {x}^{4} - 32 {x}^{3} + 24 {x}^{2} - 8 x + 1$