Find the binomial expansion of (2+x)^3
We know thst the binomial expansion formula is (1+x)n = 1+nx +(n)(n-1)(x2)/2! ...etc
Thus we want to rearrange the expression we are given into the form (1+x)n so that we can apply the formula to it.
Taking out a factor of 2 we have (2(1+(x/2)))3 which simplifies to 23(1 + (x/2) )3 = 8(1+ (x/2))3
From here we can apply the binomial expansion formula. To make the caculations simpler first we will calculate the expansion of (1+ (x/2))3 and then multiply the expansion by 8 to get our answer.
(1 + (x/2))3 = 1 + (3)(x/2) + ((3)(2)(x/2)2)/2! + ((3)(2)(1)(x/2)3)/3! = 1 + 3x/2 + 3x2/4 + x3/8
Multiplying this expansion by 8 we get 8(1 + 3x/2 + 3x2/4 + x3/8) = 8 + 12x + 6x2 + x3
Thus, (2+x)3 = 8 + 12x + 6x2 + x3
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