2021 AMC 12B Problems/Problem 20
Contents
- 1 Problem
- 2 Solution 1 (Difference of Cubes)
- 3 Solution 2 (Finds Q(z) Using Patterns)
- 4 Solution 3 (Modular Arithmetic in Polynomials)
- 5 Solution 4 (Complex Numbers)
- 6 Solution 5 (Complex Numbers but Easier)
- 7 Solution 6 (More Complex Numbers)
- 8 Solution 7 (Factorization)
- 9 Video Solution
- 10 Video Solution by OmegaLearn (Using Modular Arithmetic and Meta-Solving)
- 11 Video Solution (Long Division, Not Brutal)
- 12 See Also
Problem
Let and be the unique polynomials such thatand the degree of is less than What is
Solution 1 (Difference of Cubes)
Let be a root of so that It follows that from which but
Note that Since for each root of the remainder when is divided by is
~MRENTHUSIASM
Solution 2 (Finds Q(z) Using Patterns)
Note that the equation above is in the form of polynomial division, with being the dividend, being the divisor, and and being the quotient and remainder respectively. Since the degree of the dividend is and the degree of the divisor is , that means the degree of the quotient is . Note that can't influence the degree of the right hand side of this equation since its degree is either or . Since the coefficients of the leading term in the dividend and the divisor are both , that means the coefficient of the leading term of the quotient is also . Thus, the leading term of the quotient is . Multiplying by the divisor gives . We have our term but we have these unnecessary terms like . We can get rid of these terms by adding to the quotient to cancel out these terms, but this then gives us . Our first instinct will probably be to add , but we can't do this as although this will eliminate the term, it will produce a term. Since no other term of the form where is an integer less than will produce a term when multiplied by the divisor, we can't add to the quotient. Instead, we can add to the coefficient to get rid of the term. Continuing this pattern, we get the quotient as The last term when multiplied with the divisor gives . This will get rid of the term but will produce the expression , giving us the dividend as . Note that the dividend we want is of the form . Therefore, our remainder will have to be in order to get rid of the term in the expression and give us , which is what we want. Therefore, the remainder is
~ rohan.sp ~rocketsri
Solution 3 (Modular Arithmetic in Polynomials)
Note that so if is the remainder when dividing by , Now, So , and The answer is
Solution 4 (Complex Numbers)
One thing to note is that takes the form of for some constants and Note that the roots of are part of the solutions of They can be easily solved with roots of unity: Obviously the right two solutions are the roots of We substitute into the original equation, and becomes Using De Moivre's Theorem, we get Expanding into rectangular complex number form: Comparing the real and imaginary parts, we get So, the answer is
~Jamess2022 (burntTacos ;-))
Solution 5 (Complex Numbers but Easier)
We have motivation to get rid of the term by subtituting in either or for , as this sets equal to . Doing so, This immediately eliminates choices and as they are constants. Upon a quick check, and both don't work, as the sign of has to be negative. Checking (or realizing that it's the only one that actually works), we see that is the right answer.
-Benedict T (countmath1)
Solution 6 (More Complex Numbers)
Since we can plug in either or . Assuming we have and Solving gives us and , thus
~SirAppel
Solution 7 (Factorization)
Notice that , so Therefore, we have from which .
Video Solution
~MathProblemSolvingSkills.com
Video Solution by OmegaLearn (Using Modular Arithmetic and Meta-Solving)
~ pi_is_3.14
Video Solution (Long Division, Not Brutal)
~hippopotamus1
See Also
2021 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 19 |
Followed by Problem 21 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.