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).
A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.
Solution
The given data tells us that the roots of f(x)+x2 are −1,2,−3, and 4. Combining with the fact that f is monic and quartic we get f(x)+x2=(x+1)(x−2)(x+3)(x−4). Hence f(1)=(2)(−1)(4)(−3)−1=23.
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