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Difference between revisions of "2022 AMC 12A Problems/Problem 21"

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==Solution 3==
 
==Solution 3==
  
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Let <math>x^{1011} = u</math>, now we can rewrite our polynomial as <math>u^2+u+1</math>. Using the quadratic formula to solve for the roots of this polynomial, we have <cmath>x^{1011} = \frac{-1\pm i\sqrt{3}}{2}</cmath> Looking at our answer choices, we want to find a polynomial whose roots satisfy this expression. Since the expression <math>x^6+x^3+1</math> is in a similar form to our original polynomial, except with <math>x^3</math> in place of <math>x^{1011}</math>, this would be a good place to start. Solving for the roots of <math>x^3</math> in a similar fashion, <cmath>x^3= \frac{-1\pm i\sqrt{3}}{2}</cmath> for the solution we are testing. Now notice that we can rewrite the roots of <math>x^3</math> as <cmath>x^3 = cis{\frac{2\pi}{3}}, cis{\frac{4\pi}{3}}</cmath> Both of which are third roots of unity. We want to now check if this value of <math>x^3</math> satisfies <math>x^{1011} = \frac{-1\pm i\sqrt{3}}{2}</math>. Notice that <math>x^{1011} = (x^{3})^{112\cdot3}\cdot x^3</math>, and since both values of <math>x^3</math> are roots of unity, we can simplify the expression we want satisfiedto the expression to <math>x^{1011}=x^3</math>. Since both values of <math>x^3</math> are also values of <math>x^{1011}</math>, the roots for our <math>x^6+x^3+1</math> are also roots of <math>x^{2022}+x^{1011}+1</math>, meaning that <cmath>x^6+x^3+1 | x^{2022}+x^{1011}+1</cmath> so Therefore, the answer is <math>\boxed{E}</math>.
+
Let <math>x^{1011} = u</math>, now we can rewrite our polynomial as <math>u^2+u+1</math>. Using the quadratic formula to solve for the roots of this polynomial, we have <cmath>x^{1011} = \frac{-1\pm i\sqrt{3}}{2}</cmath> Looking at our answer choices, we want to find a polynomial whose roots satisfy this expression. Since the expression <math>x^6+x^3+1</math> is in a similar form to our original polynomial, except with <math>x^3</math> in place of <math>x^{1011}</math>, this would be a good place to start. Solving for the roots of <math>x^3</math> in a similar fashion, <cmath>x^3= \frac{-1\pm i\sqrt{3}}{2}</cmath> for the solution we are testing. Now notice that we can rewrite the roots of <math>x^3</math> as <cmath>x^3 = \operatorname{cis}{\frac{2\pi}{3}}, \operatorname{cis}{\frac{4\pi}{3}}</cmath> Both of which are third roots of unity. We want to now check if this value of <math>x^3</math> satisfies <math>x^{1011} = \frac{-1\pm i\sqrt{3}}{2}</math>. Notice that <math>x^{1011} = (x^{3})^{112\cdot3}\cdot x^3</math>, and since both values of <math>x^3</math> are roots of unity, we can simplify the expression we want satisfied to the expression to <math>x^{1011}=x^3</math>. Since both values of <math>x^3</math> are also values of <math>x^{1011}</math>, the roots for our <math>x^6+x^3+1</math> are also roots of <math>x^{2022}+x^{1011}+1</math>, meaning that <cmath>x^6+x^3+1 | x^{2022}+x^{1011}+1</cmath> so Therefore, the answer is <math>\boxed{E}</math>.
 +
 
 +
- DavidHovey
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==Solution 4 (Describe the Roots)==
 +
 
 +
We know that a monic polynomial <math>q</math> divides a monic polynomial <math>p</math> if and only if all the roots of <math>q</math> are roots of <math>p.</math> Since <cmath>P(x)=x^{2022}+x^{1011}+1=\frac{x^{3033}-1}{x^{1011}-1}</cmath>, the roots of <math>P</math> are the <math>3033</math>rd roots of unity that aren't <math>1011</math>th roots of unity.
 +
 
 +
Now, note that:
 +
 
 +
1: The roots of polynomial <math>A</math> are the primitive <math>6</math>th roots of unity.
 +
 
 +
2: The roots of polynomial <math>B</math> are the primitive cube roots of unity.
 +
 
 +
3: The roots of polynomial <math>C</math> are the primitive <math>8</math>th roots of unity.
 +
 
 +
4: The roots of polynomial <math>D</math> are the primitive <math>18</math>th roots of unity.
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 +
5: The roots of polynomial <math>E</math> are the primitive <math>9</math>th roots of unity.
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 +
However, since <math>6</math>, <math>8</math>, and <math>18</math> don't divide <math>3033</math>, the roots of polynomial <math>A</math> are not all <math>3033</math>rd roots of unity, and the same is true for polynomials <math>C</math> and <math>D</math>, eliminating choices <math>A</math>, <math>C</math> and <math>D.</math> Also, since <math>3</math> divides <math>1011</math>, the roots of polynomial <math>B</math> are all <math>1011</math>th roots of unity, eliminating choice <math>B.</math> That leaves choice <math>\boxed{E}</math>, and we can confirm that this is correct by noticing that <math>9</math> divides <math>3033</math> but not <math>1011.</math> From that, we can see that the roots of polynomial <math>E</math> are <math>3033</math>rd roots of unity but not <math>1011</math>th roots of unity, so they are all roots of <math>P.</math> Therefore, <math>E</math> divides <math>P.</math>
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~pianoboy
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==Solution 5 (Simple Elimination)==
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Put the value <math>x = -1</math>. This gives <math>P(-1) = (-1)^{2022} + (-1)^{1011} + 1 = 1</math>.
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This automatically eliminates choices <math>A</math> , <math>C</math> and <math>D</math> since they do not form a factor of <math>1</math> at <math>x = -1</math>.
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 +
 
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Solution <math>5.1</math>:
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Now, put <math>x = \omega</math> (omega) at choice <math>B</math>. We get that,
 +
<cmath>\omega^2 + \omega + 1 = 0</cmath>
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Thus choice B has <math>(x - \omega)</math> as a factor. If choice <math>B</math> were to be a factor of <math>P(x)</math>, <math>(x - \omega)</math> would also have to be a factor of <math>P(x)</math>, which is clearly not the case, as <math>P(\omega) \neq 0</math>.
 +
This eliminates choice <math>B</math>, leaving us with answer <math>\boxed{E}</math>.
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 +
 
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Solution <math>5.2</math>(Mods):
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Plug in <math>2</math> for <math>x</math> at choice <math>B</math>. <math>2^2+2*2+1=7</math>.
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The cycle of mods for <math>2</math> and <math>7</math> is <math>2</math>, <math>4</math>, <math>1</math>. Both <math>2022</math> and <math>1011</math> are divisible by <math>3</math>, so we have <math>1</math> + <math>1</math> + <math>1</math> = <math>3</math>. <math>3</math> is not divisible by <math>7</math>, so we eliminate choice <math>B</math>. This leaves us with <math>\boxed{(E) x^{6}+x^{3}+1}</math>.
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 +
 
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~SouradipClash_03
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Solution <math>5.2</math> ~ Bread 10
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===Solution 5a (Elimination but slightly different)===
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 +
Like Solution 5, let <math>x=-1</math> which eliminates the choices of <math>A</math>, <math>C</math>, and <math>D</math> as they do not divide <math>P(-1)=1</math> as they form <math>3, 2, 3</math> respectively by letting <math>x=-1</math>.
 +
 
 +
This leaves us with only <math>2</math> choices, <math>B</math> and <math>E</math>. Notice that letting <math>x=0</math> or <math>x=1</math> still make these answer choices work and the other values will leave large numbers for us to check which is not feasible in a 75 minutes math competition.
 +
 
 +
However, we notice answer choice <math>B</math> is quadratic so if answer choice <math>B</math> divides the given polynomial, then the roots of the quadratic must also be roots of the polynomial.
 +
 
 +
Through quadratic formula, we find the roots of this quadratic as <math>\dfrac{-1+i\sqrt{3}}{2}</math> and <math>\dfrac{-1-i\sqrt{3}}{2}</math>.
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 +
We notice that these roots can be written nicely in polar form <math>\cos(\dfrac{2\pi}{3})+i\sin(\dfrac{2\pi}{3})</math> or <math>\cos(\dfrac{4\pi}{3})+i\sin(\dfrac{4\pi}{3})</math>.
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We plug either one of these (preferably the root on the left as it is smaller) and see that the polynomial doesn't equal <math>0</math> suggesting that <math>B</math> is not the correct answer choice.
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As we only have one answer choice left, we choose <math>\boxed{E}</math>
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 +
~Batmanstark
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===Solution 5b (elimination but even faster and cheesier)===
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We first substitute <math>-1</math> and eliminate <math>A</math>, <math>C</math>, and <math>D</math>. We are now left with <math>B</math> and <math>E</math>. When <math>x=3</math>, <math>B=13</math>; we then calculate <math>3^{2022}+3^{1011}+1</math> mod <math>13</math>. Note that <math>2022</math> and <math>1011</math> are both multiples of <math>3</math>, and <math>3^3=27\equiv1\pmod{13}</math>. Thus, <math>3^{2022}\equiv3^{1011}\equiv3^3\equiv1\pmod{13}</math>, so it turns out that the given expression is <math>1+1+1=3</math> mod <math>13</math>. We need it to be <math>0</math>, and since <math>3\neq0</math>, then <math>B</math> is not the correct answer; <math>\boxed{\textbf{(E) }x^6+x^3+1}</math> is correct.
 +
 
 +
~Technodoggo
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 +
==Video Solution by Math-X (Smart and Simple)==
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https://youtu.be/7yAh4MtJ8a8?si=OkLQzwiqd9AnIS5K&t=6374 ~Math-X
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 +
== Video Solution==
 +
includes review of factoring polynomials
 +
 
 +
https://youtu.be/UIxePPu0Zus
 +
 
 +
~MathProblemSolvingSkills.com
  
 
== Video Solution by ThePuzzlr ==  
 
== Video Solution by ThePuzzlr ==  
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== See Also ==
 
== See Also ==
 
{{AMC12 box|year=2022|ab=A|num-b=20|num-a=22}}
 
{{AMC12 box|year=2022|ab=A|num-b=20|num-a=22}}
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[[Category:Intermediate Algebra Problems]]
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 13:00, 5 November 2024

Problem

Let \[P(x) = x^{2022} + x^{1011} + 1.\] Which of the following polynomials is a factor of $P(x)$?

$\textbf{(A)} \, x^2 -x + 1 \qquad\textbf{(B)} \, x^2 + x + 1 \qquad\textbf{(C)} \, x^4 + 1 \qquad\textbf{(D)} \, x^6 - x^3 + 1 \qquad\textbf{(E)} \, x^6 + x^3 + 1$

Solution 1

$P(x) = x^{2022} + x^{1011} + 1$ is equal to $\frac{x^{3033}-1}{x^{1011}-1}$ by difference of powers.

Therefore, the answer is a polynomial that divides $x^{3033}-1$ but not $x^{1011}-1$.

Note that any polynomial $x^m-1$ divides $x^n-1$ if and only if $m$ is a factor of $n$.

The prime factorizations of $1011$ and $3033$ are $3*337$ and $3^2*337$, respectively.

Hence, $x^9-1$ is a divisor of $x^{3033}-1$ but not $x^{1011}-1$.

By difference of powers, $x^9-1=(x^3-1)(x^6+x^3+1)$. Therefore, the answer is $\boxed{E}$.

Solution 2

We simply test roots for each, as $2022,1011$ are multiples of three, we need to make sure the roots are in the form of $e^{i\frac{k\pi}{9}}$, so we only have to look at $D,E$.

If we look at choice $E$, $x=e^{i\frac{\pm2\pi}{9}}$ which works perfectly, the answer is just $E$

~bluesoul

Solution 3

Let $x^{1011} = u$, now we can rewrite our polynomial as $u^2+u+1$. Using the quadratic formula to solve for the roots of this polynomial, we have \[x^{1011} = \frac{-1\pm i\sqrt{3}}{2}\] Looking at our answer choices, we want to find a polynomial whose roots satisfy this expression. Since the expression $x^6+x^3+1$ is in a similar form to our original polynomial, except with $x^3$ in place of $x^{1011}$, this would be a good place to start. Solving for the roots of $x^3$ in a similar fashion, \[x^3= \frac{-1\pm i\sqrt{3}}{2}\] for the solution we are testing. Now notice that we can rewrite the roots of $x^3$ as \[x^3 = \operatorname{cis}{\frac{2\pi}{3}}, \operatorname{cis}{\frac{4\pi}{3}}\] Both of which are third roots of unity. We want to now check if this value of $x^3$ satisfies $x^{1011} = \frac{-1\pm i\sqrt{3}}{2}$. Notice that $x^{1011} = (x^{3})^{112\cdot3}\cdot x^3$, and since both values of $x^3$ are roots of unity, we can simplify the expression we want satisfied to the expression to $x^{1011}=x^3$. Since both values of $x^3$ are also values of $x^{1011}$, the roots for our $x^6+x^3+1$ are also roots of $x^{2022}+x^{1011}+1$, meaning that \[x^6+x^3+1 | x^{2022}+x^{1011}+1\] so Therefore, the answer is $\boxed{E}$.

- DavidHovey

Solution 4 (Describe the Roots)

We know that a monic polynomial $q$ divides a monic polynomial $p$ if and only if all the roots of $q$ are roots of $p.$ Since \[P(x)=x^{2022}+x^{1011}+1=\frac{x^{3033}-1}{x^{1011}-1}\], the roots of $P$ are the $3033$rd roots of unity that aren't $1011$th roots of unity.

Now, note that:

1: The roots of polynomial $A$ are the primitive $6$th roots of unity.

2: The roots of polynomial $B$ are the primitive cube roots of unity.

3: The roots of polynomial $C$ are the primitive $8$th roots of unity.

4: The roots of polynomial $D$ are the primitive $18$th roots of unity.

5: The roots of polynomial $E$ are the primitive $9$th roots of unity.

However, since $6$, $8$, and $18$ don't divide $3033$, the roots of polynomial $A$ are not all $3033$rd roots of unity, and the same is true for polynomials $C$ and $D$, eliminating choices $A$, $C$ and $D.$ Also, since $3$ divides $1011$, the roots of polynomial $B$ are all $1011$th roots of unity, eliminating choice $B.$ That leaves choice $\boxed{E}$, and we can confirm that this is correct by noticing that $9$ divides $3033$ but not $1011.$ From that, we can see that the roots of polynomial $E$ are $3033$rd roots of unity but not $1011$th roots of unity, so they are all roots of $P.$ Therefore, $E$ divides $P.$

~pianoboy

Solution 5 (Simple Elimination)

Put the value $x = -1$. This gives $P(-1) = (-1)^{2022} + (-1)^{1011} + 1 = 1$. This automatically eliminates choices $A$ , $C$ and $D$ since they do not form a factor of $1$ at $x = -1$.


Solution $5.1$: Now, put $x = \omega$ (omega) at choice $B$. We get that, \[\omega^2 + \omega + 1 = 0\] Thus choice B has $(x - \omega)$ as a factor. If choice $B$ were to be a factor of $P(x)$, $(x - \omega)$ would also have to be a factor of $P(x)$, which is clearly not the case, as $P(\omega) \neq 0$. This eliminates choice $B$, leaving us with answer $\boxed{E}$.


Solution $5.2$(Mods): Plug in $2$ for $x$ at choice $B$. $2^2+2*2+1=7$. The cycle of mods for $2$ and $7$ is $2$, $4$, $1$. Both $2022$ and $1011$ are divisible by $3$, so we have $1$ + $1$ + $1$ = $3$. $3$ is not divisible by $7$, so we eliminate choice $B$. This leaves us with $\boxed{(E) x^{6}+x^{3}+1}$.


~SouradipClash_03

Solution $5.2$ ~ Bread 10

Solution 5a (Elimination but slightly different)

Like Solution 5, let $x=-1$ which eliminates the choices of $A$, $C$, and $D$ as they do not divide $P(-1)=1$ as they form $3, 2, 3$ respectively by letting $x=-1$.

This leaves us with only $2$ choices, $B$ and $E$. Notice that letting $x=0$ or $x=1$ still make these answer choices work and the other values will leave large numbers for us to check which is not feasible in a 75 minutes math competition.

However, we notice answer choice $B$ is quadratic so if answer choice $B$ divides the given polynomial, then the roots of the quadratic must also be roots of the polynomial.

Through quadratic formula, we find the roots of this quadratic as $\dfrac{-1+i\sqrt{3}}{2}$ and $\dfrac{-1-i\sqrt{3}}{2}$.

We notice that these roots can be written nicely in polar form $\cos(\dfrac{2\pi}{3})+i\sin(\dfrac{2\pi}{3})$ or $\cos(\dfrac{4\pi}{3})+i\sin(\dfrac{4\pi}{3})$.

We plug either one of these (preferably the root on the left as it is smaller) and see that the polynomial doesn't equal $0$ suggesting that $B$ is not the correct answer choice.

As we only have one answer choice left, we choose $\boxed{E}$

~Batmanstark

Solution 5b (elimination but even faster and cheesier)

We first substitute $-1$ and eliminate $A$, $C$, and $D$. We are now left with $B$ and $E$. When $x=3$, $B=13$; we then calculate $3^{2022}+3^{1011}+1$ mod $13$. Note that $2022$ and $1011$ are both multiples of $3$, and $3^3=27\equiv1\pmod{13}$. Thus, $3^{2022}\equiv3^{1011}\equiv3^3\equiv1\pmod{13}$, so it turns out that the given expression is $1+1+1=3$ mod $13$. We need it to be $0$, and since $3\neq0$, then $B$ is not the correct answer; $\boxed{\textbf{(E) }x^6+x^3+1}$ is correct.

~Technodoggo

Video Solution by Math-X (Smart and Simple)

https://youtu.be/7yAh4MtJ8a8?si=OkLQzwiqd9AnIS5K&t=6374 ~Math-X

Video Solution

includes review of factoring polynomials

https://youtu.be/UIxePPu0Zus

~MathProblemSolvingSkills.com

Video Solution by ThePuzzlr

https://youtu.be/YRcaIrwA2AU

~ MathIsChess

See Also

2022 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 20 		Followed by
Problem 22
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

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