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Difference between revisions of "2003 AIME II Problems/Problem 9"

 
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== Problem ==
 
== Problem ==
 +
Consider the polynomials <math>P(x) = x^{6} - x^{5} - x^{3} - x^{2} - x</math> and <math>Q(x) = x^{4} - x^{3} - x^{2} - 1.</math> Given that <math>z_{1},z_{2},z_{3},</math> and <math>z_{4}</math> are the roots of <math>Q(x) = 0,</math> find <math>P(z_{1}) + P(z_{2}) + P(z_{3}) + P(z_{4}).</math>
  
 
== Solution ==
 
== Solution ==
 +
When we use long division to divide <math>P(x)</math> by <math>Q(x)</math>, the remainder is <math>x^2-x+1</math>.
 +
 +
So, since <math>z_1</math> is a root, <math>P(z_1)=(z_1)^2-z_1+1</math>.
 +
 +
Now this also follows for all roots of <math>Q(x)</math>
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Now <cmath>P(z_2)+P(z_1)+P(z_3)+P(z_4)=z_1^2-z_1+1+z_2^2-z_2+1+z_3^2-z_3+1+z_4^2-z_4+1</cmath>
 +
 +
Now by [[Vieta's formulas|Vieta's]] we know that <math>-z_4-z_3-z_2-z_1=-1</math>,
 +
so by [[Newton's Sums]] we can find <math>z_1^2+z_2^2+z_3^2+z_4^2</math>
 +
 +
<math>a_ns_2+a_{n-1}s_1+2a_{n-2}=0</math>
 +
 +
<math>(1)(s_2)+(-1)(1)+2(-1)=0</math>
 +
 +
<math>s_2-1-2=0</math>
 +
 +
<math>s_2=3</math>
 +
 +
So finally
 +
<math>P(z_2)+P(z_1)+P(z_3)+P(z_4)=3+4-1=\boxed{006}.</math>
 +
 +
== Solution 2 ==
 +
 +
Let <math>S_k=z_1^k+z_2^k+z_3^k+z_4^k</math> then by [[Vieta's Formula]] we have
 +
<cmath>S_{-1}=\frac{z_1z_2z_3+z_1z_3z_4+z_1z_2z_4+z_1z_2z_3}{z_1z_2z_3z_4}=0</cmath>
 +
<cmath>S_0=4</cmath>
 +
<cmath>S_1=1</cmath>
 +
<cmath>S_2=3</cmath>
 +
By [[Newton's Sums]] we have
 +
<cmath>a_4S_k+a_3S_{k-1}+a_2S_{k-2}+a_1S_{k-1}+a_0S_{k-4}=0</cmath>
 +
 +
Applying the formula couples of times yields <math>P(z_1)+P(z_2)+P(z_3)+P(z_4)=S_6-S_5-S_3-S_2-S_1=\boxed{6}</math>.
 +
 +
~ Nafer
 +
 +
== Solution 3 ==
 +
 +
<math>P(x) = x^{2}Q(x)+x^{4}-x^{3}-x.</math>
 +
 +
So we just have to find: <math>\sum_{n=1}^{4} z^{4}_n - \sum_{n=1}^{4} z^{3}_n - \sum_{n=1}^{4} z_n</math>.
 +
 +
And by [[Newton's Sums]] this computes to: <math>11-4-1 = \boxed{006}</math>.
 +
 +
~ LuisFonseca123
 +
 +
== Solution 4 ==
 +
If we scale <math>Q(x)</math> by <math>x^2</math>, we get <math>x^6-x^5-x^4-x^2</math>. In order to get to <math>P(x)</math>, we add <math>x^4-x^3-x</math>. Therefore, our answer is <math>\sum_{n=1}^{4} z^4_n-z^3_n-z_n</math>. However, rearranging <math>Q(z_n) = 0</math>, makes our final answer <math>\sum_{n=1}^{4} z^2_n-z_n+1</math>. The sum of the squares of the roots is <math>1^2-2(-1) = 3</math> and the sum of the roots is <math>1</math>. Adding 4 to our sum, we get <math>3-1+4 = \boxed{006}</math>.
 +
 +
~ Vedoral
 +
 +
== Solution 5 ==
 +
Let <math>S_k</math> = <math>z_1^k+z_2^k+z_3^k+z_4^k</math>
 +
 +
By [[Newton's Sums]],
 +
 +
<math>S_1-1=0</math>
 +
 +
<math>S_2-S_1-2=0</math>
 +
 +
<math>S_3-S_2-S_1=0</math>
 +
 +
<math>S_4-S_3-S_2-4=0</math>
 +
 +
<math>S_5-S_4-S_3-S_1=0</math>
 +
 +
<math>S_6-S_5-S_4-S_2=0</math>
 +
 +
Solving for <math>S_1,S_2,S_3,S_4,S_5,S_6</math>, we get <math>S_1=1, S_2=3, S_3=4, S_4=11, S_5=16, S_6=30</math>
 +
 +
<math>P(z_1)+P(z_2)+P(z_3)+P(z_4)=S_6-S_5-S_3-S_2-S_1=\boxed{006}</math>
 +
 +
== Video Solution by Sal Khan ==
 +
https://www.youtube.com/watch?v=ZSESJ8TeGSI&list=PLSQl0a2vh4HCtW1EiNlfW_YoNAA38D0l4&index=14
 +
- AMBRIGGS
 +
 +
 +
[rule]
 +
 +
Nice!-sleepypuppy
  
 
== See also ==
 
== See also ==
−
* [[2003 AIME II Problems]]
+
{{AIME box|year=2003|n=II|num-b=8|num-a=10}}
 +
 
 +
[[Category: Intermediate Algebra Problems]]
 +
{{MAA Notice}}

Latest revision as of 01:41, 11 September 2024

Problem

Consider the polynomials $P(x) = x^{6} - x^{5} - x^{3} - x^{2} - x$ and $Q(x) = x^{4} - x^{3} - x^{2} - 1.$ Given that $z_{1},z_{2},z_{3},$ and $z_{4}$ are the roots of $Q(x) = 0,$ find $P(z_{1}) + P(z_{2}) + P(z_{3}) + P(z_{4}).$

Solution

When we use long division to divide $P(x)$ by $Q(x)$, the remainder is $x^2-x+1$.

So, since $z_1$ is a root, $P(z_1)=(z_1)^2-z_1+1$.

Now this also follows for all roots of $Q(x)$ Now \[P(z_2)+P(z_1)+P(z_3)+P(z_4)=z_1^2-z_1+1+z_2^2-z_2+1+z_3^2-z_3+1+z_4^2-z_4+1\]

Now by Vieta's we know that $-z_4-z_3-z_2-z_1=-1$, so by Newton's Sums we can find $z_1^2+z_2^2+z_3^2+z_4^2$

$a_ns_2+a_{n-1}s_1+2a_{n-2}=0$

$(1)(s_2)+(-1)(1)+2(-1)=0$

$s_2-1-2=0$

$s_2=3$

So finally $P(z_2)+P(z_1)+P(z_3)+P(z_4)=3+4-1=\boxed{006}.$

Solution 2

Let $S_k=z_1^k+z_2^k+z_3^k+z_4^k$ then by Vieta's Formula we have \[S_{-1}=\frac{z_1z_2z_3+z_1z_3z_4+z_1z_2z_4+z_1z_2z_3}{z_1z_2z_3z_4}=0\] \[S_0=4\] \[S_1=1\] \[S_2=3\] By Newton's Sums we have \[a_4S_k+a_3S_{k-1}+a_2S_{k-2}+a_1S_{k-1}+a_0S_{k-4}=0\]

Applying the formula couples of times yields $P(z_1)+P(z_2)+P(z_3)+P(z_4)=S_6-S_5-S_3-S_2-S_1=\boxed{6}$.

~ Nafer

Solution 3

$P(x) = x^{2}Q(x)+x^{4}-x^{3}-x.$

So we just have to find: $\sum_{n=1}^{4} z^{4}_n - \sum_{n=1}^{4} z^{3}_n - \sum_{n=1}^{4} z_n$.

And by Newton's Sums this computes to: $11-4-1 = \boxed{006}$.

~ LuisFonseca123

Solution 4

If we scale $Q(x)$ by $x^2$, we get $x^6-x^5-x^4-x^2$. In order to get to $P(x)$, we add $x^4-x^3-x$. Therefore, our answer is $\sum_{n=1}^{4} z^4_n-z^3_n-z_n$. However, rearranging $Q(z_n) = 0$, makes our final answer $\sum_{n=1}^{4} z^2_n-z_n+1$. The sum of the squares of the roots is $1^2-2(-1) = 3$ and the sum of the roots is $1$. Adding 4 to our sum, we get $3-1+4 = \boxed{006}$.

~ Vedoral

Solution 5

Let $S_k$ = $z_1^k+z_2^k+z_3^k+z_4^k$

By Newton's Sums,

$S_1-1=0$

$S_2-S_1-2=0$

$S_3-S_2-S_1=0$

$S_4-S_3-S_2-4=0$

$S_5-S_4-S_3-S_1=0$

$S_6-S_5-S_4-S_2=0$

Solving for $S_1,S_2,S_3,S_4,S_5,S_6$, we get $S_1=1, S_2=3, S_3=4, S_4=11, S_5=16, S_6=30$

$P(z_1)+P(z_2)+P(z_3)+P(z_4)=S_6-S_5-S_3-S_2-S_1=\boxed{006}$

Video Solution by Sal Khan

https://www.youtube.com/watch?v=ZSESJ8TeGSI&list=PLSQl0a2vh4HCtW1EiNlfW_YoNAA38D0l4&index=14 - AMBRIGGS


[rule]

Nice!-sleepypuppy

See also

2003 AIME II (Problems • Answer Key • Resources)
Preceded by
Problem 8 		Followed by
Problem 10
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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