How do you use the binomial series to expand # (x - 2)^9#?

Question

Madelyn Anderson · Accepted Answer

#x^9 -18x^8 + 144x^7 -672x^6 + 2016x^5 -4032x^4 +5376x^3 -4608^2 + 2304x -512#

#9#th row of pascal's triangle:

1., 9., 36, 84, 126, 126, 84, 36, 9, 1

the powers of each term in the binomial expression always add to #9:# #(x^9)(-2^0)+(x^8)(-2^1)+(x^7)(-2^2)+(x^6)(-2^3)+(x^5)(-2^4)+(x^4)(-2^5)+(x^3)(-2^6)+(x^2)(-2^7)+(x^1)(-2^8)+(x^0)(-2^9)#

This is what results from multiplying each term by the matching coefficient:

#1(x^9)(-2^0)+ 9(x^8)(-2^1)+ 36(x^7)(-2^2)+ 84(x^6)(-2^3)+ 126(x^5)(-2^4)+ 126(x^4)(-2^5)+ 84(x^3)(-2^6)+ 36(x^2)(-2^7)+ 9(x^1)(-2^8)+1(x^0)(-2^9)#

#n^0=1#

#(x^9)+ 9(x^8)((-2)^1)+ 36(x^7)((-2)^2)+ 84(x^6)((-2)^3)+ 126(x^5)((-2)^4)+ 126(x^4)((-2)^5)+ 84(x^3)((-2)^6)+ 36(x^2)((-2)^7)+ 9(x^1)((-2)^8)+((-2)^9)#

#=x^9+ 9x^8(-2)^1+ 36(x^7)((-2)^2)+ 84(x^6)((-2)^3)+ 126(x^5)((-2)^4)+ 126(x^4)(-2)^5+ 84(x^3)((-2)^6)+ 36(x^2)((-2)^7)+ 9(x^1)((-2)^8)+((-2)^9)#

#=x^9 -18x^8 + 144x^7 -672x^6 + 2016x^5 -4032x^4 +5376x^3 -4608^2 + 2304x -512#