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(C) The sum of the coefficients of a polynomial $P(x)$ is obtained by evaluating $P(1)$.Given the polynomial $P(x) = (\alpha ^2 x^2 - 2\alpha x + 1)^{

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(C) The sum of the coefficients of a polynomial $P(x)$ is obtained by evaluating $P(1)$.Given the polynomial $P(x) = (\alpha ^2 x^2 - 2\alpha x + 1)^{51}$.Setting $x = 1$,the sum of the coefficients is $(\alpha ^2(1)^2 - 2\alpha(1) + 1)^{51} = (\alpha ^2 - 2\alpha + 1)^{51}$.Since the sum of the coefficients vanishes,we have $(\alpha ^2 - 2\alpha + 1)^{51} = 0$.This implies $(\alpha - 1)^2 = 0$,which gives $\alpha = 1$.

If the sum of the coefficients in the expansion of $({\alpha ^2}{x^2} - 2\alpha x + 1)^{51}$ vanishes,then the value of $\alpha$ is

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