An algebra problem by Ilham Saiful Fauzi
Algebra
Level
5
Find the least positive integer n such that the polynomial can be written as product of four non-constant polynomials with integer coefficients
The answer is 16.
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1 solution
Pick n = 1 6 then factor the polynomial x 1 2 + 6 4 using the "sum of cubes" technique.
Result is a product of 4 factors: ( x 4 + 2 x 3 + 2 x 2 + 4 x + 4 ) ( x 4 − 2 x 3 + 2 x 2 − 4 x + 4 ) ( x 2 + 2 x + 2 ) ( x 2 − 2 x + 2 )
Reasoning: We need 4 n to be a number which can be written as a product of 2 natural numbers which are themselves, cubes. Forgetting 1, the smallest cube is 2 3 , so 8 × 8 = 6 4 and n = 1 6 making the degree of the polynomial to factor: 1 2