How do I use Pascal's triangle to expand #(x - 1)^5#?
1 Answer
The answer is:
When expanding, we consider the general form:
Recall that the first row of Pascal's Triangle is:
#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#
