(y-4)^3(y−4)3 is simplified.
However, if you want to expand it using binomial expansion, you can use Pascal's triangle:
Pascal's triangle gives the combinations:
" "1 1 " "(a + b)^0 (a+b)0
" "1 " " 1" 1 1 " "(a + b)^1 (a+b)1
" "1 " "2" " 1" 1 2 1 " "(a + b)^2 (a+b)2
" "1 " " 3 " " 3 " "1 1 3 3 1 " "(a + b)^3 (a+b)3
Binomial Theorem: (a + b)^n = (a+b)n=
""_nC_0 a^n b^0 + _nC_1 a^(n-1) b^1 + _nC_2a^(n-2)b^2 + ... + _nCn a^0 b^n
where ""_nC_0 = (n!)/((n-0!)(0!)) = 1; " " _nC_1 = (n!)/((n-1!)(1!))
""_nC_2 = (n!)/((n-2!)(2!)); " "_nC_n = (n!)/((n-n!)(n!)) = 1
Given: simplify (y-4)^3
Let a = y " and " b = -4 ; " " n =3
From Pascal's triangle: ""_3C_0 = ""_3C_3 = 1; ""_3C_1 = ""_3C_2 = 3
(y-4)^3 =
""_3C_0 y^3 (-4)^0 + ""_3C_1 y^2 (-4)^1 + ""_3C_2 y^1 (-4)^2 + ""_3C_3 y^0 (-4)^3 =
y^3 + 3y^2(-4) + 3y(16) -64 =
y^3 -12y^2 +48y - 64
