Problem 16 · 2017 Math Kangaroo
Hard
Number Theory
divisibility
The polynomial \(5x^3 + ax^2 + bx + 24\) has whole-number coefficients a and b. Which of the following numbers is definitely not a solution to the equation \(5x^3 + ax^2 + bx + 24 = 0\)?
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Answer: D — 5
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Hint 1 of 2
Any integer root of a polynomial with integer coefficients must divide the constant term.
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Hint 2 of 2
The constant term is 24; which listed candidate is not a divisor of 24?
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Approach: rational (integer) root test on the constant term
- An integer solution r of 5x^3 + ax^2 + bx + 24 = 0 must divide the constant term 24.
- Among 1, -1, 3, 5, 6, all divide 24 except 5.
- So 5 can never be a solution, whatever a and b are.
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