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Difference between revisions of "2010 AMC 12A Problems/Problem 21"

(Created page with '== Problem 21 == The graph of <math>y=x^6-10x^5+29x^4-4x^3+ax^2</math> lies above the line <math>y=bx+c</math> except at three values of <math>x</math>, where the graph and the l…')
 
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== Problem 21 ==
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== Problem ==
 
The graph of <math>y=x^6-10x^5+29x^4-4x^3+ax^2</math> lies above the line <math>y=bx+c</math> except at three values of <math>x</math>, where the graph and the line intersect. What is the largest of these values?
 
The graph of <math>y=x^6-10x^5+29x^4-4x^3+ax^2</math> lies above the line <math>y=bx+c</math> except at three values of <math>x</math>, where the graph and the line intersect. What is the largest of these values?
  
 
<math>\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 8</math>
 
<math>\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 8</math>
  
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[[2010 AMC 12A Problems/Problem 21|Solution]]
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== Solutions ==
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=== Solution 1 ===
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== Solution ==
 
 
The <math>x</math> values in which <math>y=x^6-10x^5+29x^4-4x^3+ax^2</math> intersect at <math>y=bx+c</math> are the same as the zeros of <math>y=x^6-10x^5+29x^4-4x^3+ax^2-bx-c</math>.
 
The <math>x</math> values in which <math>y=x^6-10x^5+29x^4-4x^3+ax^2</math> intersect at <math>y=bx+c</math> are the same as the zeros of <math>y=x^6-10x^5+29x^4-4x^3+ax^2-bx-c</math>.
  
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Suppose we let <math>p</math>, <math>q</math>, and <math>r</math> be the roots of this function, and let <math>x^3-ux^2+vx-w</math> be the cubic polynomial with roots <math>p</math>, <math>q</math>, and <math>r</math>.
 
Suppose we let <math>p</math>, <math>q</math>, and <math>r</math> be the roots of this function, and let <math>x^3-ux^2+vx-w</math> be the cubic polynomial with roots <math>p</math>, <math>q</math>, and <math>r</math>.
  
 +
<cmath>\begin{align*}(x-p)(x-q)(x-r) &= x^3-ux^2+vx-w\\
 +
(x-p)^2(x-q)^2(x-r)^2 &= x^6-10x^5+29x^4-4x^3+ax^2-bx-c = 0\\
 +
\sqrt{x^6-10x^5+29x^4-4x^3+ax^2-bx-c} &= x^3-ux^2+vx-w = 0\end{align*}</cmath>
 +
 +
In order to find <math>\sqrt{x^6-10x^5+29x^4-4x^3+ax^2-bx-c}</math> we must first expand out the terms of <math>(x^3-ux^2+vx-w)^2</math>.
 +
 +
<cmath>(x^3-ux^2+vx-w)^2</cmath>
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<cmath>= x^6-2ux^5+(u^2+2v)x^4-(2uv+2w)x^3+(2uw+v^2)x^2-2vwx+w^2</cmath>
 +
 +
[Quick note: Since we don't know <math>a</math>, <math>b</math>, and <math>c</math>, we really don't even need the last 3 terms of the expansion.]
 +
 +
<cmath>\begin{align*}&2u = 10\\
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u^2+2v &= 29\\
 +
2uv+2w &= 4\\
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u &= 5\\
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v &= 2\\
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w &= -8\\
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&\sqrt{x^6-10x^5+29x^4-4x^3+ax^2-bx-c} = x^3-5x^2+2x+8\end{align*}</cmath>
 +
 +
All that's left is to find the largest root of <math>x^3-5x^2+2x+8</math>.
 +
 +
<cmath>\begin{align*}&x^3-5x^2+2x+8 = (x-4)(x-2)(x+1)\\
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&\boxed{\textbf{(A)}\ 4}\end{align*}</cmath>
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 +
=== Solution 2 ===
 +
The <math>x</math> values in which <math>y=x^6-10x^5+29x^4-4x^3+ax^2</math> intersect at <math>y=bx+c</math> are the same as the zeros of <math>y=x^6-10x^5+29x^4-4x^3+ax^2-bx-c</math>.
 +
We also know that this graph has 3 places tangent to the x-axis, which means that each root has to have a multiplicity of 2.
 +
Let the function be <math>(x-p)^2(x-q)^2(x-r)^2</math>.
 +
 +
Applying Vieta's formulas, we get <math>2p+2q+2r = 10</math>  or  <math>p+q+r = 5</math>.
 +
Applying it again, we get, after simplification, <math>p^2+q^2+r^2+4pq+4pr+4qr = 29</math>.
  
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<math>(x-p)(x-q)(x-r) = x^3-ux^2+vx-w</math>
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Notice that squaring the first equation yields <math>p^2+q^2+r^2+2pq+2qr+2pr= 25</math>, which is similar to the second equation.
  
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<math>(x-p)^2(x-q)^2(x-r)^2 = x^6-10x^5+29x^4-4x^3+ax^2-bx-c = 0</math>
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Subtracting this from the second equation, we get <math>2pq+2pr+2qr = 4</math>. Now that we have the <math>pq+pr+qr</math> term, we can manpulate the equations to
 +
yield the sum of squares. <math>2(p^2+q^2+r^2+2pq+2qr+2pr)-2pq-2pr-2qr= 25*2-4</math> or  <math>2p^2+2q^2+2r^2+2pq+2qr+2pr = 46</math>. We finally reach <math>(p+q)^2+(q+r)^2+(p+r)^2 = 46</math>.
  
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<math>\sqrt{x^6-10x^5+29x^4-4x^3+ax^2-bx-c} = x^3-ux^2+vx-w = 0</math>
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Since the answer choices are integers, we can guess and check squares to get <math>\{(p+q)^2, (q+r)^2, (p+r)^2\} = \{1, 9, 36\}</math> in some order. We can check that this works by adding then and seeing <math>2p+2q+2r = 10</math>. We just need to take the lowest value in the set, square root it, and subtract the resulting value from 5 to get <math>\boxed{\textbf{(A)}\ 4}</math>.
  
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Note: One could also multiply <math>2pq+2pr+2qr = 4</math> by 2 and subtract from <math>p^2+q^2+r^2+4pq+4pr+4qr = 29</math> to obtain <cmath>p^2+q^2+r^2=21</cmath> The ordered triple {16,4,1} sums to 21, and the answer choices are all positive integers, therefore the answer is 4.
  
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In order to find <math>\sqrt{x^6-10x^5+29x^4-4x^3+ax^2-bx-c}</math> we must first expand out the terms of <math>(x^3-ux^2+vx-w)^2</math>.
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==== Alternative method:====
  
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After reaching <math>p+q+r = 5</math> and <math>pq + qr + rp = 2</math>, we can algebraically derive <math>pqr</math>.
  
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<math>(x^3-ux^2+vx-w)^2</math>
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Applying Vieta's formulas on the <math>x^3</math> term yields <math>2p^2q+2pq^2+2q^2r+2qr^2+2r^2p+2rp^2+8pqr = 4</math>.
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<math>= x^6-2ux^5+(u^2+2v)x^4-(2uv+2w)x^3+(2uw+v^2)x^2-2vwx+w^2</math>
 
  
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[Quick note: Since we don't know <math>a</math>, <math>b</math>, and <math>c</math>, we really don't even need the last 3 terms of the expansion.]
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Notice that <math>(p+q+r)(pq+qr+rp) = p^2q+pq^2+q^2r+qr^2+r^2p+rp^2+3pqr</math>, so <cmath>2p^2q+2pq^2+2q^2r+2qr^2+2r^2p+2rp^2+6pqr = 2(p+q+r)(pq+qr+rp) = 20.</cmath>
  
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<math>2u = 10</math>
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Subtracting this from <math>2p^2q+2pq^2+2q^2r+2qr^2+2r^2p+2rp^2+8pqr = 4</math> yields <math>2pqr = -16</math>, so <math>pqr = -8</math>, which means that <math>p</math>, <math>q</math>, and <math>r</math> are the roots of the cubic <math>x^3 - 5x^2 + 2x + 8</math>, and it is not hard to find that these roots are <math>-1</math>, <math>2</math>, and <math>4</math>. The largest of these values is <math>\boxed{\textbf{(A)}\ 4}</math>.
  
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<math>u^2+2v = 29</math>
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Extra note: One can also use the identity
 +
<cmath>(p+q)(q+r)(p+r)=2pqr+\sum_{sym}p^2q</cmath>
 +
to compute <math>pqr</math>. From <math>p^2q+pq^2+q^2r+qr^2+r^2p+rp^2+4pqr = 2</math>, use our identity to get <math>2pqr+(p+q)(q+r)(p+r)=2</math>. Then use <math>p+q+r=5</math> to rewrite as <math>2pqr+(5-p)(5-q)(5-r)=2</math>. Expanding and using <math>pq+qr+pr=2</math> as well gives the result <math>pqr=-8</math>.
  
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<math>2uv+2w = 4</math>
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~~ clarkculus
  
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<math>u = 5</math>
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== Solution 3==
 +
First, <math>y=x^6-10x^5+29x^4-4x^3+ax^2-bx-c = 0</math> has exactly <math>3</math> roots. Therefore, <math>y = (kx^3+lx^2+mx+n)^2 = 0</math>.
  
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<math>v = 2</math>
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So, <math>k^2x^6+2klx^5+(2km+l^2)x^4+2(kn+lm)x^3+ax^2-bx-c = 0</math>
  
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<math>w = -8</math>
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By matching the coefficients of the first <math>4</math> terms, we have <math>k^2 = 1, 2kl = -10, 2km+l^2 = 29, 2kn+2lm = -4</math>
  
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<math>\sqrt{x^6-10x^5+29x^4-4x^3+ax^2-bx-c} = x^3-5x^2+2x+8</math>
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Solving the equations above, we have <math>2</math> sets of solutions; first set of which is <math>k = 1, l = -5, m = 2, n = 8</math>. Second set of which is <math>k = -1, l = 5, m = -2, n = -8</math>. After squaring both sets, they are the same i.e. <math>x^3-5x^2+2x+8 = 0</math>.
  
 +
This is equal to <math>(x-4)(x-2)(x+1) = 0</math>. The largest root is <math>\boxed {\textbf{(A) 4}}</math>
  
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All that's left is to find the largest root of <math>x^3-5x^2+2x+8</math>.
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~Arcticturn
  
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<math>x^3-5x^2+2x+8 = (x-4)(x-2)(x+1)</math>
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== See also ==
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{{AMC12 box|year=2010|num-b=20|num-a=22|ab=A}}
  
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<math>\boxed{\textbf{(A)}\ 4}</math>
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[[Category:Intermediate Algebra Problems]]
 +
{{MAA Notice}}

Latest revision as of 16:45, 26 August 2024

Problem

The graph of $y=x^6-10x^5+29x^4-4x^3+ax^2$ lies above the line $y=bx+c$ except at three values of $x$, where the graph and the line intersect. What is the largest of these values?

$\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 8$

Solutions

Solution 1

The $x$ values in which $y=x^6-10x^5+29x^4-4x^3+ax^2$ intersect at $y=bx+c$ are the same as the zeros of $y=x^6-10x^5+29x^4-4x^3+ax^2-bx-c$.

Since there are $3$ zeros and the function is never negative, all $3$ zeros must be double roots because the function's degree is $6$.

Suppose we let $p$, $q$, and $r$ be the roots of this function, and let $x^3-ux^2+vx-w$ be the cubic polynomial with roots $p$, $q$, and $r$.

\begin{align*}(x-p)(x-q)(x-r) &= x^3-ux^2+vx-w\\ (x-p)^2(x-q)^2(x-r)^2 &= x^6-10x^5+29x^4-4x^3+ax^2-bx-c = 0\\ \sqrt{x^6-10x^5+29x^4-4x^3+ax^2-bx-c} &= x^3-ux^2+vx-w = 0\end{align*}

In order to find $\sqrt{x^6-10x^5+29x^4-4x^3+ax^2-bx-c}$ we must first expand out the terms of $(x^3-ux^2+vx-w)^2$.

\[(x^3-ux^2+vx-w)^2\] \[= x^6-2ux^5+(u^2+2v)x^4-(2uv+2w)x^3+(2uw+v^2)x^2-2vwx+w^2\]

[Quick note: Since we don't know $a$, $b$, and $c$, we really don't even need the last 3 terms of the expansion.]

\begin{align*}&2u = 10\\ u^2+2v &= 29\\ 2uv+2w &= 4\\ u &= 5\\ v &= 2\\ w &= -8\\ &\sqrt{x^6-10x^5+29x^4-4x^3+ax^2-bx-c} = x^3-5x^2+2x+8\end{align*}

All that's left is to find the largest root of $x^3-5x^2+2x+8$.

\begin{align*}&x^3-5x^2+2x+8 = (x-4)(x-2)(x+1)\\ &\boxed{\textbf{(A)}\ 4}\end{align*}

Solution 2

The $x$ values in which $y=x^6-10x^5+29x^4-4x^3+ax^2$ intersect at $y=bx+c$ are the same as the zeros of $y=x^6-10x^5+29x^4-4x^3+ax^2-bx-c$. We also know that this graph has 3 places tangent to the x-axis, which means that each root has to have a multiplicity of 2. Let the function be $(x-p)^2(x-q)^2(x-r)^2$.

Applying Vieta's formulas, we get $2p+2q+2r = 10$ or $p+q+r = 5$. Applying it again, we get, after simplification, $p^2+q^2+r^2+4pq+4pr+4qr = 29$.

Notice that squaring the first equation yields $p^2+q^2+r^2+2pq+2qr+2pr= 25$, which is similar to the second equation.

Subtracting this from the second equation, we get $2pq+2pr+2qr = 4$. Now that we have the $pq+pr+qr$ term, we can manpulate the equations to yield the sum of squares. $2(p^2+q^2+r^2+2pq+2qr+2pr)-2pq-2pr-2qr= 25*2-4$ or $2p^2+2q^2+2r^2+2pq+2qr+2pr = 46$. We finally reach $(p+q)^2+(q+r)^2+(p+r)^2 = 46$.

Since the answer choices are integers, we can guess and check squares to get $\{(p+q)^2, (q+r)^2, (p+r)^2\} = \{1, 9, 36\}$ in some order. We can check that this works by adding then and seeing $2p+2q+2r = 10$. We just need to take the lowest value in the set, square root it, and subtract the resulting value from 5 to get $\boxed{\textbf{(A)}\ 4}$.

Note: One could also multiply $2pq+2pr+2qr = 4$ by 2 and subtract from $p^2+q^2+r^2+4pq+4pr+4qr = 29$ to obtain \[p^2+q^2+r^2=21\] The ordered triple {16,4,1} sums to 21, and the answer choices are all positive integers, therefore the answer is 4.

Alternative method:

After reaching $p+q+r = 5$ and $pq + qr + rp = 2$, we can algebraically derive $pqr$.

Applying Vieta's formulas on the $x^3$ term yields $2p^2q+2pq^2+2q^2r+2qr^2+2r^2p+2rp^2+8pqr = 4$.

Notice that $(p+q+r)(pq+qr+rp) = p^2q+pq^2+q^2r+qr^2+r^2p+rp^2+3pqr$, so \[2p^2q+2pq^2+2q^2r+2qr^2+2r^2p+2rp^2+6pqr = 2(p+q+r)(pq+qr+rp) = 20.\]

Subtracting this from $2p^2q+2pq^2+2q^2r+2qr^2+2r^2p+2rp^2+8pqr = 4$ yields $2pqr = -16$, so $pqr = -8$, which means that $p$, $q$, and $r$ are the roots of the cubic $x^3 - 5x^2 + 2x + 8$, and it is not hard to find that these roots are $-1$, $2$, and $4$. The largest of these values is $\boxed{\textbf{(A)}\ 4}$.

Extra note: One can also use the identity \[(p+q)(q+r)(p+r)=2pqr+\sum_{sym}p^2q\] to compute $pqr$. From $p^2q+pq^2+q^2r+qr^2+r^2p+rp^2+4pqr = 2$, use our identity to get $2pqr+(p+q)(q+r)(p+r)=2$. Then use $p+q+r=5$ to rewrite as $2pqr+(5-p)(5-q)(5-r)=2$. Expanding and using $pq+qr+pr=2$ as well gives the result $pqr=-8$.

~~ clarkculus

Solution 3

First, $y=x^6-10x^5+29x^4-4x^3+ax^2-bx-c = 0$ has exactly $3$ roots. Therefore, $y = (kx^3+lx^2+mx+n)^2 = 0$.

So, $k^2x^6+2klx^5+(2km+l^2)x^4+2(kn+lm)x^3+ax^2-bx-c = 0$

By matching the coefficients of the first $4$ terms, we have $k^2 = 1, 2kl = -10, 2km+l^2 = 29, 2kn+2lm = -4$

Solving the equations above, we have $2$ sets of solutions; first set of which is $k = 1, l = -5, m = 2, n = 8$. Second set of which is $k = -1, l = 5, m = -2, n = -8$. After squaring both sets, they are the same i.e. $x^3-5x^2+2x+8 = 0$.

This is equal to $(x-4)(x-2)(x+1) = 0$. The largest root is $\boxed {\textbf{(A) 4}}$

~Arcticturn

See also

2010 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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