Sum Of Coefficients

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If every variable in the polynomial expansion of (3x - 4y)^{12} is set equal to one, the resulting sum of all the coefficients of the expansion equals which value?

RankGuru · Accepted Answer

The sum of all coefficients of any polynomial expression is obtained by substituting one for every variable, a quick evaluation trick central to JEE Advanced binomial problems. For (3x - 4y)^{12}, setting x = 1 and y = 1 gives (3·1 - 4·1)^{12} = (3 - 4)^{12} = (-1)^{12}. Since the exponent 12 is even, (-1)^{12} = 1, so the sum of all coefficients is exactly 1. Option -1 would arise only for an odd exponent, which is not the case here. Option 7^{12} mistakenly adds the magnitudes 3 and 4 instead of subtracting, ignoring the sign in the binomial. Option 0 incorrectly assumes the positive and negative coefficients cancel completely, which they do not. The key insight is that substituting unity collapses the entire structure into a single power of the numerical base. Plausibility check: since the base 3 - 4 = -1 has magnitude one, any power of it must have magnitude one, and the even exponent fixes the sign as positive, fully consistent with the answer 1.

Sum Of Coefficients

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