Difference between revisions of "2021 AMC 12B Problems/Problem 20"

Revision as of 15:54, 13 February 2021

Problem

Let $Q(z)$ and $R(z)$ be the unique polynomials such that $z^{2021}+1=(z^2+z+1)Q(z)+R(z)$ and the degree of $R$ is less than $2.$ What is $R(z)?$

$\textbf{(A) }-z \qquad \textbf{(B) }-1 \qquad \textbf{(C) }2021\qquad \textbf{(D) }z+1 \qquad \textbf{(E) }2z+1$

Video Solution using long division(not brutal)

https://youtu.be/kxPDeQRGLEg ~hippopotamus1

Solution 1

Note that $z^3-1\equiv 0\pmod{z^2+z+1}$ so if $F(z)$ is the remainder when dividing by $z^3-1$ , $F(z)\equiv R(z)\pmod{z^2+z+1}.$ Now, $z^{2021}+1= (z^3-1)(z^{2018} + z^{2015} + \cdots + z^2) + z^2+1$ So $F(z) = z^2+1$ , and $R(z)\equiv F(z) \equiv -z\pmod{z^2+z+1}$ The answer is $\boxed{\textbf{(A) }-z}.$

Solution 1b (More Thorough Version of 1)

Instead of dealing with a nasty $z^2+z+1$ , we can instead deal with the nice $z^3 - 1$ , as $z^2+z+1$ is a factor of $z^3-1$ . Then, we try to see what $\frac{z^{2021} + 1}{z^3 - 1}$ is. Of course, we will need a $z^{2018}$ , getting $z^{2021} - z^{2018}$ . Then, we've gotta get rid of the $z^{2018}$ term, so we add a $z^{2015}$ , to get $z^{2021} - z^{2015}$ . This pattern continues, until we add a $z^2$ to get rid of $z^5$ , and end up with $z^{2021} - z^2$ . We can't add anything more to get rid of the $z^2$ , so our factor is $z^{2018} + z^{2015} + z^{2012} + \cdots + z^2$ . Then, to get rid of the $z^2$ , we must have a remainder of $+z^2$ , and to get the $+1$ we have to also have a $+1$ in the remainder. So, our product is $z^{2021}+1= (z^3-1)(z^{2018} + z^{2015} + \cdots + z^2) + z^2+1.$ Then, our remainder is $z^2+1$ . The remainder when dividing by $z^3-1$ must be the same when dividing by $z^2+z+1$ , modulo $z^2+z+1$ . So, we have that $z^2 + 1 \equiv R(z) \pmod{z^2+z+1}$ , or $R(z) \equiv -z\pmod{z^2+z+1}$ . This corresponds to answer choice $\boxed{\textbf{(A)} ~ -z}$ . ~rocketsri

Solution 2 (Complex numbers)

One thing to note is that $R(z)$ takes the form of $Az + B$ for some constants A and B. Note that the roots of $z^2 + z + 1$ are part of the solutions of $z^3 -1 = 0$ They can be easily solved with roots of unity: $z^3 = 1$ $z^3 = e^{i 0}$ $z = e^{i 0}, e^{i \frac{2\pi}{3}}, e^{i -\frac{2\pi}{3}}$ $\newline$ Obviously the right two solutions are the roots of $z^2 + z + 1 = 0$ We substitute $e^{i \frac{2\pi}{3}}$ into the original equation, and $z^2 + z + 1$ becomes 0. Using De Moivre's theorem, we get: $e^{i\frac{4042\pi}{3}} + 1 = A \cdot e^{i \frac{2\pi}{3}} + B$ $e^{i\frac{4\pi}{3}} + 1 = A \cdot e^{i \frac{2\pi}{3}} + B$ Expanding into rectangular complex number form: $\frac{1}{2} - \frac{\sqrt{3}}{2} i = (-\frac{1}{2}A + B) + \frac{\sqrt{3}}{2} i A$ Comparing the real and imaginary parts, we get: $A = -1, B = 0$ The answer is $\boxed{\textbf{(A) }-z}$ . ~Jamess2022(burntTacos;-;)

Solution 3(similar to Solution 1b)

Note that the equation above is in the form of polynomial division, with $z^{2021}+1$ being the dividend, $z^2+z+1$ being the divisor, and $Q(x)$ and $R(x)$ being the quotient and remainder respectively. Since the degree of the dividend is $2021$ and the degree of the divisor is $2$ , that means the degree of the quotient is $2021-2 = 2019$ . Note that R(x) can't influence the degree of the right hand side of this equation since its degree is either $1$ or $0$ . Since the coefficients of the leading term in the dividend and the divisor are both $1$ , that means the coefficient of the leading term of the quotient is also $1$ . Thus, the leading term of the quotient is $z^{2019}$ . Multiplying $z^{2019}$ by the divisor gives $z^{2021}+z^{2020}+z^{2019}$ . We have our $z^{2021}$ term but we have these unnecessary terms like $z^{2020}$ . We can get rid of these terms by adding $-z^{2018}$ to the quotient to cancel out these terms, but this then gives us $z^{2021}-z^{2018}$ . Our first instinct will probably be to add $z^{2017}$ , but we can't do this as although this will eliminate the $-z^{2018}$ term, it will produce a $z^{2019}$ term. Since no other term of the form $z^n$ where $n$ is an integer less than $2017$ will produce a $z^{2019}$ term when multiplied by the divisor, we can't add $z^{2017}$ to the quotient. Instead, we can add $z^{2016}$ to the coefficient to get rid of the $-z^{2018}$ term. Continuing this pattern, we get the quotient as $z^{2019}-z^{2018}+z^{2016}-z^{2015}+....-z^2+1.$ The last term when multiplied with the divisor gives $z^2+z+1$ . This will get rid of the $-z^2$ term but will produce the expression $z+1$ , giving us the dividend as $z^{2021}+z+1$ . Note that the dividend we want is of the form $z^{2021}+1$ . Therefore, our remainder will have to be $-z$ in order to get rid of the $z$ term in the expression and give us $z^{2021}+1$ , which is what we want. Therefore, the remainder is $\boxed{\textbf{(A) }-z \qquad}$

~ rohan.sp

Video Solution by OmegaLearn (Using Modular Arithmetic and Meta-solving)

https://youtu.be/nnjr17q7fS0

~ pi_is_3.14

@@ Line 41: / Line 41: @@
 The answer is <math>\boxed{\textbf{(A) }-z}</math>. ~Jamess2022(burntTacos;-;)
-Solution 3(similar to Solution 1b):
+==Solution 3(similar to Solution 1b)==
 Note that the equation above is in the form of polynomial division, with <math>z^{2021}+1</math> being the dividend, <math>z^2+z+1</math> being the divisor, and <math>Q(x)</math> and <math>R(x)</math> being the quotient and remainder respectively. Since the degree of the dividend is <math>2021</math> and the degree of the divisor is <math>2</math>, that means the degree of the quotient is <math>2021-2 = 2019</math>. Note that R(x) can't influence the degree of the right hand side of this equation since its degree is either <math>1</math> or <math>0</math>. Since the coefficients of the leading term in the dividend and the divisor are both <math>1</math>, that means the coefficient of the leading term of the quotient is also <math>1</math>. Thus, the leading term of the quotient is <math>z^{2019}</math>. Multiplying <math>z^{2019}</math> by the divisor gives <math>z^{2021}+z^{2020}+z^{2019}</math>. We have our <math>z^{2021}</math> term but we have these unnecessary terms like <math>z^{2020}</math>. We can get rid of these terms by adding <math>-z^{2018}</math> to the quotient to cancel out these terms, but this then gives us <math>z^{2021}-z^{2018}</math>. Our first instinct will probably be to add <math>z^{2017}</math>, but we can't do this as although this will eliminate the <math>-z^{2018}</math> term, it will produce a <math>z^{2019}</math> term. Since no other term of the form <math>z^n</math> where <math>n</math> is an integer less than <math>2017</math> will produce a <math>z^{2019}</math> term when multiplied by the divisor, we can't add <math>z^{2017}</math> to the quotient. Instead, we can add <math>z^{2016}</math> to the coefficient to get rid of the <math>-z^{2018}</math> term. Continuing this pattern, we get the quotient as <cmath>z^{2019}-z^{2018}+z^{2016}-z^{2015}+....-z^2+1.</cmath>
 The last term when multiplied with the divisor gives <math>z^2+z+1</math>. This will get rid of the <math>-z^2</math> term but will produce the expression <math>z+1</math>, giving us the dividend as <math>z^{2021}+z+1</math>. Note that the dividend we want is of the form <math>z^{2021}+1</math>. Therefore, our remainder will have to be <math>-z</math> in order to get rid of the <math>z</math> term in the expression and give us <math>z^{2021}+1</math>, which is what we want. Therefore, the remainder is <math>\boxed{\textbf{(A) }-z \qquad}</math>
+~ rohan.sp
 == Video Solution by OmegaLearn (Using Modular Arithmetic and Meta-solving) ==

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
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