If f(x) is a monic quartic polynomial such that f(−1)=−1, f(2)=−4, f(−3)=−9, and f(4)=−16, find f(1).
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Let g(x)=f(x)+x2. Then g(x) is also a monic quartic polynomial, and g(−1)=g(2)=g(−3)=f(4)=0, so g(x)=(x+1)(x−2)(x+3)(x−4).Hence, f(x)=(x+1)(x−2)(x+3)(x−4)−x2. In particular, f(1)=(2)(−1)(4)(−3)−1=23.