Question

How do I use Pascal's triangle to expand `(x - 1)^5`?

Answer 1

The answer is: `x^5-5x^4+10x^3-10x^2+5x-1`

When expanding, we consider the general form: `(x+y)^n`.

Recall that the first row of Pascal's Triangle is: `(x+y)^0`. So for `(x-1)^5`, we are looking at the `6^(th)` row of Pascal's Triangle for the coefficients:

`color(white)((color(black)((,,,,,1,,,,,),(,,,,1,,1,,,,),(,,,1,,2,,1,,,),(,,1,,3,,3,,1,,),(,1,,4,,6,,4,,1,),(color(red)1,,color(blue)5,,color(green)10,,color(orange)10,,color(olive)5,,color(pink)1)))`

Expanding, we get:

`color(red)1*x^5y^0+color(blue)5*x^4y^1+color(green)10*x^3y^2+color(orange)10*x^2y^3+color(olive)5*x^1y^4+color(pink)1*x^0y^5`

Now we substitute and simplify:

`x^5+5x^4(-1)^1+10*^3(-1)^2+10x^2(-1)^3+5x^1(-1)^4+(-1)^5`
`=x^5-5x^4+10x^3-10x^2+5x-1`