Come posso usare il triangolo di Pascal per espandere # (x - 1) ^ 5 #?

La risposta è: #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)))#

In espansione, otteniamo:

#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#