2021 AIME II Problems/Problem 4
There are real numbers and such that is a root of and is a root of These two polynomials share a complex root where and are positive integers and Find
Solution 1 (Complex Conjugate Root Theorem)
By the Complex Conjugate Root Theorem, the imaginary roots for each of and are a pair of complex conjugates. Let and It follows that the roots of are and the roots of are
We know that
Applying Vieta's Formulas to we have from which
Applying Vieta's Formulas to we have from which Finally, we get by and
Solution 2 (Somewhat Bashy)
Also, , hence
satisfies both we can put it in both equations and equate to 0.
In the first equation, we get Simplifying this further, we get
In the second equation, we get Simplifying this further, we get
Comparing (1) and (2),
Substituting these in gives,
This simplifies to
Consider case of :
(because c = 1) Also, Also, Equation (2) gives
Solving (4) and (5) simultaneously gives
[AIME can not have more than one answer, so we can stop here also 😁... Not suitable for Subjective exam]
Solution 3 (Heavy Calculation Solution)
start off by applying vieta's and you will find that and . After that, we have to use the fact that and are roots of and , respectively. Since we know that if you substitute the root of a function back into the function, the output is zero, therefore and and you can set these two equations equal to each other while also substituting the values of , , , and above to give you , then you can rearrange the equation into . With this property, we know that is divisible by therefore that means which results in which finally gives us m=10 mod 21. We can test the first obvious value of which is and we see that this works as we get and . That means your answer will be
Solution 4 (Synthetic Division)
We note that and for some polynomials and . Through synthetic division (ignoring the remainder as we can set and to constant values such that the remainder is zero), , and By the complex conjugate root theorem, we know that and share the same roots, and they share the same leading coefficient, so . Therefore, and . Solving the system equations, we get and , so . Finally, by the quadratic formula, we have roots of , so our final answer is
Solution 5 (Fast and Easy)
We plug -20 into the equation obtaining , likewise, plugging -21 into the second equation gets .
Both equations must have 3 solutions exactly, so the other two solutions must be and .
By Vieta's, the sum of the roots in the first equation is , so must be 10.
Next, using Vieta's theorem on the second equation, you get: However, since we know that the sum of the roots with complex numbers are 20, we can factor out the terms with -21, so
Given that is , then is equal to .
Therefore, the answer to the equation is
Video Solution by TheCALT
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