2010 AMC 10B Problems/Problem 25
Contents
Problem
Let , and let be a polynomial with integer coefficients such that
, and
.
What is the smallest possible value of ?
Solution
We observe that because , if we define a new polynomial such that , has roots when ; namely, when .
Thus since has roots when , we can factor the product out of to obtain a new polynomial such that .
Then, plugging in values of we get
Thus, the least value of must be the . Solving, we receive , so our answer is .
Critique
The above solution is incomplete. What is really proven is that 315 is a factor of , if such an exists. That only rules out answer A.
To prove that the answer is correct, one could exhibit a polynomium that satisfies the requirements with . Here's one: .
You get that from the matrix and and computing which comes out as the all-integer coefficients above.
Critique of the Critique
Once you find that is a factor of 315, you can instantly find that the answer is 315 because the problem asks for the smallest value of .
Critique of the (Critique of the Critique)
First of all, the solution shows that is a multiple of , not a factor of . Many people confuse the usage of the words 'factor' and 'multiple'. Secondly, even if is a multiple of , it does not mean that you can instantly get that the answer is because we need to know that is possible. After all, is also a multiple of , but is definitely not the smallest possible number.
To complete the solution, we can let , and then try to find . We know from the above calculation that , and . Then we can let , getting . Let , then . Therefore, it is possible to choose , so the goal is accomplished. As a reference, the polynomial we get is
See also
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