# How do you expand the binomial (x-y)^5?

May 18, 2018

The expansion is $- {y}^{5} + 5 {y}^{4} x - 10 {y}^{3} {x}^{2} + 10 {y}^{4} {x}^{3} - 5 {y}^{5} {x}^{4} + {x}^{5}$.

#### Explanation:

We need to use Pascal's Triangle, shown in the picture below, for this expansion.

Because the binomial is raised to the $5 t h$ power, we need to use the $5 t h$ row of the triangle. The $5 t h$ row is the one that features $\textcolor{red}{1 , 5 , 10 , 10 , 5 ,}$ and $\textcolor{red}{1}$.

In the expansion, $\textcolor{n a v y}{x}$ will be the first term and $\textcolor{g r e e n}{- y}$ will be the second. Thus, the expression looks like this.

(color(red)1*color(green)((-y)^5)*color(navy)(x^0))+(color(red)5*color(green)((-y)^4)*color(navy)(x^1))+(color(red)10*color(green)((-y)^3)*color(navy)(x^2))+ (color(red)10*color(green)((-y)^4)*color(navy)(x^3))+ (color(red)5*color(green)((-y)^5)*color(navy)(x^4))+(color(red)1*color(green)((-y)^0)*color(navy)(x^5))

For each term from Pascal's Triangle, the exponent of the first term, $\textcolor{n a v y}{x}$, increases by $1$, while the exponent of the second term, $\textcolor{g r e e n}{- y}$, decreases by $1$.

Now, we can simplify and combine like terms.

$\textcolor{g r e e n}{- {y}^{5}} + \textcolor{red}{5} \textcolor{g r e e n}{{y}^{4} \textcolor{n a v y}{x}} - \textcolor{red}{10} \textcolor{g r e e n}{{y}^{3}} \textcolor{n a v y}{{x}^{2}} + \textcolor{red}{10} \textcolor{g r e e n}{{y}^{4}} \textcolor{n a v y}{{x}^{3}} - \textcolor{red}{5} \textcolor{g r e e n}{{y}^{5}} \textcolor{n a v y}{{x}^{4}} + \textcolor{n a v y}{{x}^{5}}$