It is known that p(x) is a monic polynomial of degree 5. Moreover, it is known that the solutions of the equation p(x)=0 are exactly x=0,1,2,4. Determine the maximum value that the coefficient of the first-degree term can take.
Note: a polynomial is monic if the coefficient of its highest degree term (in our case: the fifth-degree one) is 1.
Pick one
Solution
The answer is (C). The polynomial can be written as the product of five first-degree factors: x(x−1)(x−2)(x−4)(x−k), where k∈{0,1,2,4}. The first-degree term has coefficient (−1)(−2)(−4)(−k)=8k, which is maximal for k=4 and in that case equals 32.
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Source: MathNet,
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