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Let f(x) be a quartic polynomial with integer coefficients and four integer roots. Suppose the constant term of f(x) is 6 .

(a) Is it possible for x=3 to be a root of f(x)?

(b) Is it possible for x=3 to be a double root of f(x) ?

Apr 30, 2019

#1
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$$\text{all the roots are integers so we have}\\ f(x) = (x-i_1)(x-i_2)(x-i_3)(x-i_4)\\ \text{The constant term is }c_0 = i_1 i_2 i_3 i_4 = 6\\ \text{3 can be a root as }3\cdot 2 = 6\\ \text{On the other hand 3 cannot be a double root as }3\cdot 3 = 9\\ \text{and there is no combination of integer factors that will multiply 9 to obtain 6}$$

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May 1, 2019
#2
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Thank you this response is short and to the point while still helping me understand the problem

May 2, 2019