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Problem 640

AMC 10/12, early questions

Algebra Difficulty 3.8 Multiple choice Slovenia — National Math Olympiad · Slovenia · 2012

How many polynomials of degree $5$ with the coefficients being $1$ or $-1$ have a root at $1$ ?

Pick one

Official solution

The value of the polynomial at $1$ is equal to the sum of all the coefficients of the polynomial. Since this value must be $0$ , exactly three coefficients must be $1$ and three must be $-1$ . The number of such polynomials is thus $\binom{6}{3} = 20$ .

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