A non-zero polynomial ݂\(f(x)\) of degree 3 has roots at \(x = 1 , x = 2\) and…
2014
A non-zero polynomial ݂\(f(x)\) of degree 3 has roots at \(x = 1 , x = 2\) and \(x = 3\) Which one of the following must be TRUE?
- A.
\(f(0)f(4)< 0\) - B.
\(f(0)f(4)> 0\) - C.
\(f(0)+f(4)> 0\) - D.
\(f(0)+f(4)< 0\)
Attempted by 39 students.
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Correct answer: A
Since f(x) is a non-zero cubic with roots at x = 1, 2, 3, it must be of the form f(x) = a(x-1)(x-2)(x-3) with a ≠ 0.
Evaluate at 0: f(0) = a(-1)(-2)(-3) = -6a.
Evaluate at 4: f(4) = a(3)(2)(1) = 6a.
Product: f(0)f(4) = (-6a)(6a) = -36a2, which is negative for any nonzero a, so f(0)f(4) < 0.
Sum: f(0) + f(4) = -6a + 6a = 0, so statements claiming the sum is > 0 or < 0 are false.
Conclusion: The only statement that must be true is that f(0)f(4) < 0.