Question

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

Answer 1

The expansion is `-y^5+5y^4x-10y^3x^2+10y^4x^3-5y^5x^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 `5th` power, we need to use the `5th` row of the triangle. The `5th` row is the one that features `color(red) (1,5,10,10,5,)` and `color(red)1`.

In the expansion, `color(navy)x` will be the first term and `color(green)(-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, `color(navy)x`, increases by `1`, while the exponent of the second term, `color(green)(-y)`, decreases by `1`.

Now, we can simplify and combine like terms.

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