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Algebra Difficulty 3.8 AMC 10/12 Find the answer Slovenia

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

Pick one

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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