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Difference between revisions of "2005 AIME I Problems/Problem 6"

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== Problem ==
 
== Problem ==
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Let <math> P </math> be the [[product]] of the [[nonreal]] [[root]]s of <math> x^4-4x^3+6x^2-4x=2005. </math> Find <math> \lfloor P\rfloor. </math>
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Let <math> P </math> be the product of the nonreal roots of <math> x^4-4x^3+6x^2-4x=2005. </math> Find <math> \lfloor P\rfloor. </math>
  
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== Solution ==
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== Solution 1 ==
 
The left-hand side of that [[equation]] is nearly equal to <math>(x - 1)^4</math>.  Thus, we add 1 to each side in order to complete the fourth power and get
 
The left-hand side of that [[equation]] is nearly equal to <math>(x - 1)^4</math>.  Thus, we add 1 to each side in order to complete the fourth power and get
 
<math>(x - 1)^4 = 2006</math>.
 
<math>(x - 1)^4 = 2006</math>.
  
−
Let <math>r = \sqrt[4]{2006}</math> be the positive [[real]] fourth root of 2006.  Then the roots of the above equation are <math>x = 1 + i^n r</math> for <math>n = 0, 1, 2, 3</math>.  The two non-real members of this set are <math>1 + ir</math> and <math>1 - ir</math>.  Their product is <math>\displaystyle P = 1 + r^2 = 1 + \sqrt{2006}</math>.  <math>44^2 = 1936 < 2006 < 2025 = 45^2</math> so <math>\lfloor P \rfloor = 1 + 44 = 045</math>.
+
Let <math>r = \sqrt[4]{2006}</math> be the positive [[real]] fourth root of 2006.  Then the roots of the above equation are <math>x = 1 + i^n r</math> for <math>n = 0, 1, 2, 3</math>.  The two non-real members of this set are <math>1 + ir</math> and <math>1 - ir</math>.  Their product is <math>P = 1 + r^2 = 1 + \sqrt{2006}</math>.  <math>44^2 = 1936 < 2006 < 2025 = 45^2</math> so <math>\lfloor P \rfloor = 1 + 44 = \boxed{045}</math>.
 +
 
 +
== Solution 2 ==
 +
 
 +
Starting like before,
 +
<math>(x-1)^4= 2006</math>
 +
This time we apply differences of squares.
 +
<math>(x-1)^4-2006=0</math> so
 +
<math>((x-1)^2+\sqrt{2006})((x-1)^2 -\sqrt{2006})=0</math>
 +
If you think of each part of the product as a quadratic, then <math>((x-1)^2+\sqrt{2006})</math> is bound to hold the two  non-real roots since the other definitely crosses the x-axis twice since it is just <math>x^2</math> translated down and right.
 +
Therefore <math>P</math> is the product of the roots of <math>((x-1)^2+\sqrt{2006})</math> or <math> P=1+\sqrt{2006}</math> so
 +
 
 +
<math>\lfloor P \rfloor = 1 + 44 = \boxed{045}</math>.
 +
 
 +
== Solution 3 ==
 +
 
 +
If we don't see the fourth power, we can always factor the LHS to try to create a quadratic substitution. Checking, we find that <math>x=0</math> and <math>x=2</math> are both roots. Synthetic division gives
 +
<math>(x^2-2x)(x^2-2x+2)=2005</math>. We now have our quadratic substitution of
 +
<math>y=x^2-2x+1=(x-1)^2</math>, giving us
 +
<math>(y-1)(y+1)=2005</math>. From here we proceed as in Solution 1 to get <math>\boxed{045}</math>.
 +
 
 +
== Solution 4 ==
 +
 
 +
Realizing that if we add 1 to both sides we get <math>x^4-4x^3+6x^2-4x+1=2006</math> which can be factored as <math>(x-1)^4=2006</math>. Then we can substitute <math>(x-1)</math> with <math>y</math> which leaves us with <math>y^4=2006</math>. Now subtracting 2006 from both sides we get some difference of squares <math>y^4-2006=0 \rightarrow (y-\sqrt[4]{2006})(y+\sqrt[4]{2006})(y^2+\sqrt{2006})=0</math>. The question asks for the product of the complex roots so we only care about the last factor which is equal to zero. From there we can solve <math>y^2+\sqrt{2006}=0</math>, we can substitute <math>(x-1)</math> for <math>y</math> giving us  <math>(x-1)^2+\sqrt{2006}=0</math>, expanding this we get <math>x^2-2x+1+\sqrt{2006}=0</math>. We know that the product of a quadratics roots is <math>\frac{c}{a}</math> which leaves us with <math>\frac{1+\sqrt{2006}}{1}=1+\sqrt{2006}\approx\boxed{045}</math>.
 +
 
 +
== Solution 5 ==
 +
 
 +
As in solution 1, we find that <math>(x-1)^4 = 2006</math>. Now <math>x-1=\pm \sqrt[4]{2006}</math> so <math>x_1 = 1+\sqrt[4]{2006}</math> and <math>x_2 = 1-\sqrt[4]{2006}</math> are the real roots of the equation. Multiplying, we get <math>x_1 x_2 = 1 - \sqrt{2006}</math>. Now transforming the original function and using Vieta's formula, <math>x^4-4x^3+6x^2-4x-2005=0</math> so <math>x_1 x_2 x_3 x_4 = \frac{-2005}{1} = -2005</math>. We find that the product of the nonreal roots is <math>x_3 x_4 = \frac{-2005}{1-\sqrt{2006}} \approx 45.8</math> and we get <math>\boxed{045}</math>.
 +
 
 +
 
 +
 
 +
Note: <math>\frac{2005}{\sqrt[4]{2006}-1}=\frac{2005(1+\sqrt[4]{2006})}{2005} = 1+\sqrt[4]{2006}.</math>
 +
 
 +
==Solution 6 (De Moivre's Theorem)==
 +
 
 +
As all the other solutions, we find that <math>(x-1)^4 = 2006</math>. Thus <math>x=\sqrt[4]{2006}+1</math>. Thus <math>x= \sqrt[4]{2006}(\cos(\frac{2\pi(k)}{4}+i\sin(\frac{2\pi(k)}{4}))+1</math> when <math>k=0,1,2,3</math>. The complex values of <math>x</math> are the ones where <math>i\sin(\frac{2\pi(k)}{4})</math> does not equal 0. These complex roots are <math>1+\sqrt[4]{2006}(i)</math> and <math>1-\sqrt[4]{2006}(i)</math>. The product of these two nonreal roots is (<math>1+\sqrt[4]{2006}(i)</math>)(<math>1-\sqrt[4]{2006}(i)</math>) which is equal to <math>1+\sqrt {2006}</math>. The floor of that value is <math>\boxed{045}</math>.
 +
 
  
  
 
== See also ==
 
== See also ==
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* [[2005 AIME I Problems/Problem 5 | Previous problem]]
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Video Solution
−
* [[2005 AIME I Problems/Problem 7 | Next problem]]
+
https://www.youtube.com/watch?v=LbHg1Su2Rmg
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* [[2005 AIME I Problems]]
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{{AIME box|year=2005|n=I|num-b=5|num-a=7}}
  
 
[[Category:Intermediate Algebra Problems]]
 
[[Category:Intermediate Algebra Problems]]
−
[[Category:Intermediate Complex Numbers Problems]]
+
{{MAA Notice}}

Latest revision as of 11:52, 13 June 2024

Problem

Let $P$ be the product of the nonreal roots of $x^4-4x^3+6x^2-4x=2005.$ Find $\lfloor P\rfloor.$

Solution 1

The left-hand side of that equation is nearly equal to $(x - 1)^4$. Thus, we add 1 to each side in order to complete the fourth power and get $(x - 1)^4 = 2006$.

Let $r = \sqrt[4]{2006}$ be the positive real fourth root of 2006. Then the roots of the above equation are $x = 1 + i^n r$ for $n = 0, 1, 2, 3$. The two non-real members of this set are $1 + ir$ and $1 - ir$. Their product is $P = 1 + r^2 = 1 + \sqrt{2006}$. $44^2 = 1936 < 2006 < 2025 = 45^2$ so $\lfloor P \rfloor = 1 + 44 = \boxed{045}$.

Solution 2

Starting like before, $(x-1)^4= 2006$ This time we apply differences of squares. $(x-1)^4-2006=0$ so $((x-1)^2+\sqrt{2006})((x-1)^2 -\sqrt{2006})=0$ If you think of each part of the product as a quadratic, then $((x-1)^2+\sqrt{2006})$ is bound to hold the two non-real roots since the other definitely crosses the x-axis twice since it is just $x^2$ translated down and right. Therefore $P$ is the product of the roots of $((x-1)^2+\sqrt{2006})$ or $P=1+\sqrt{2006}$ so

$\lfloor P \rfloor = 1 + 44 = \boxed{045}$.

Solution 3

If we don't see the fourth power, we can always factor the LHS to try to create a quadratic substitution. Checking, we find that $x=0$ and $x=2$ are both roots. Synthetic division gives $(x^2-2x)(x^2-2x+2)=2005$. We now have our quadratic substitution of $y=x^2-2x+1=(x-1)^2$, giving us $(y-1)(y+1)=2005$. From here we proceed as in Solution 1 to get $\boxed{045}$.

Solution 4

Realizing that if we add 1 to both sides we get $x^4-4x^3+6x^2-4x+1=2006$ which can be factored as $(x-1)^4=2006$. Then we can substitute $(x-1)$ with $y$ which leaves us with $y^4=2006$. Now subtracting 2006 from both sides we get some difference of squares $y^4-2006=0 \rightarrow (y-\sqrt[4]{2006})(y+\sqrt[4]{2006})(y^2+\sqrt{2006})=0$. The question asks for the product of the complex roots so we only care about the last factor which is equal to zero. From there we can solve $y^2+\sqrt{2006}=0$, we can substitute $(x-1)$ for $y$ giving us $(x-1)^2+\sqrt{2006}=0$, expanding this we get $x^2-2x+1+\sqrt{2006}=0$. We know that the product of a quadratics roots is $\frac{c}{a}$ which leaves us with $\frac{1+\sqrt{2006}}{1}=1+\sqrt{2006}\approx\boxed{045}$.

Solution 5

As in solution 1, we find that $(x-1)^4 = 2006$. Now $x-1=\pm \sqrt[4]{2006}$ so $x_1 = 1+\sqrt[4]{2006}$ and $x_2 = 1-\sqrt[4]{2006}$ are the real roots of the equation. Multiplying, we get $x_1 x_2 = 1 - \sqrt{2006}$. Now transforming the original function and using Vieta's formula, $x^4-4x^3+6x^2-4x-2005=0$ so $x_1 x_2 x_3 x_4 = \frac{-2005}{1} = -2005$. We find that the product of the nonreal roots is $x_3 x_4 = \frac{-2005}{1-\sqrt{2006}} \approx 45.8$ and we get $\boxed{045}$.


Note: $\frac{2005}{\sqrt[4]{2006}-1}=\frac{2005(1+\sqrt[4]{2006})}{2005} = 1+\sqrt[4]{2006}.$

Solution 6 (De Moivre's Theorem)

As all the other solutions, we find that $(x-1)^4 = 2006$. Thus $x=\sqrt[4]{2006}+1$. Thus $x= \sqrt[4]{2006}(\cos(\frac{2\pi(k)}{4}+i\sin(\frac{2\pi(k)}{4}))+1$ when $k=0,1,2,3$. The complex values of $x$ are the ones where $i\sin(\frac{2\pi(k)}{4})$ does not equal 0. These complex roots are $1+\sqrt[4]{2006}(i)$ and $1-\sqrt[4]{2006}(i)$. The product of these two nonreal roots is ($1+\sqrt[4]{2006}(i)$)($1-\sqrt[4]{2006}(i)$) which is equal to $1+\sqrt {2006}$. The floor of that value is $\boxed{045}$.


See also

Video Solution https://www.youtube.com/watch?v=LbHg1Su2Rmg

2005 AIME I (Problems • Answer Key • Resources)
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