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Difference between revisions of "2013 AIME I Problems/Problem 5"

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==Problem 5==
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== Problem ==
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The real root of the equation <math>8x^3 - 3x^2 - 3x - 1 = 0</math> can be written in the form <math>\frac{\sqrt[3]a + \sqrt[3]b + 1}{c}</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are positive integers. Find <math>a+b+c</math>.
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__TOC__
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== Solutions ==
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=== Solution 1 ===
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We have that <math>9x^3 = (x+1)^3</math>, so it follows that <math>\sqrt[3]{9}x = x+1</math>. Solving for <math>x</math> yields <math>\frac{1}{\sqrt[3]{9}-1} = \frac{\sqrt[3]{81}+\sqrt[3]{9}+1}{8}</math>, so the answer is <math>\boxed{98}</math>.
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=== Solution 2 ===
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Let <math>r</math> be the real root of the given [[polynomial]]. Now define the cubic polynomial <math>Q(x)=-x^3-3x^2-3x+8</math>. Note that <math>1/r</math> must be a root of <math>Q</math>. However we can simplify <math>Q</math> as <math>Q(x)=9-(x+1)^3</math>, so we must have that <math>(\frac{1}{r}+1)^3=9</math>. Thus <math>\frac{1}{r}=\sqrt[3]{9}-1</math>, and <math>r=\frac{1}{\sqrt[3]{9}-1}</math>. We can then multiply the numerator and denominator of <math>r</math> by <math>\sqrt[3]{81}+\sqrt[3]{9}+1</math> to rationalize the denominator, and we therefore have <math>r=\frac{\sqrt[3]{81}+\sqrt[3]{9}+1}{8}</math>, and the answer is <math>\boxed{98}</math>.
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=== Solution 3 ===
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It is clear that for the algebraic degree of <math>x</math> to be <math>3</math> that there exists some cubefree integer <math>p</math> and positive integers <math>m,n</math> such that <math>a = m^3p</math> and <math>b = n^3p^2</math> (it is possible that <math>b = n^3p</math>, but then the problem wouldn't ask for both an <math>a</math> and <math>b</math>). Let <math>f_1</math> be the [[automorpism]] over <math>\mathbb{Q}[\sqrt[3]{a}][\omega]</math> which sends <math>\sqrt[3]{a} \to \omega \sqrt[3]{a}</math> and <math>f_2</math> which sends <math>\sqrt[3]{a} \to \omega^2 \sqrt[3]{a}</math>  (note : <math>\omega</math> is a cubic [[root of unity]]).
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Letting <math>r</math> be the root, we clearly we have <math>r + f_1(r) + f_2(r) = \frac{3}{8}</math> by Vieta's. Thus it follows <math>c=8</math>.
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Now, note that <math>\sqrt[3]{a} + \sqrt[3]{b} + 1</math> is a root of <math>x^3 - 3x^2 - 24x - 64 = 0</math>. Thus <math>(x-1)^3 = 27x + 63</math> so <math>(\sqrt[3]{a} + \sqrt[3]{b})^3 = 27(\sqrt[3]{a} + \sqrt[3]{b}) + 90</math>. Checking the non-cubicroot dimension part, we get <math>a + b = 90</math> so it follows that <math>a + b + c = \boxed{98}</math>.
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=== Solution 4 ===
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We proceed by using the [[cubic formula]].
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Let <math>a=8</math>, <math>b=-3</math>, <math>c=-3</math>, and <math>d=-1</math>. Then let <math>m=\left(\dfrac{-b^3}{27a^3}+\dfrac{bc}{6a^2}-\dfrac{d}{2a}\right)</math> and <math>n=\left(\dfrac{c}{3a}-\dfrac{b^2}{9a^2}\right)</math>. Then the real root of <math>ax^3+bx^2+cx+d</math> is
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<cmath>\sqrt[3]{m+\sqrt{m^2+n^3}}+\sqrt[3]{m-\sqrt{m^2+n^3}}-\dfrac{b}{3a}</cmath>
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Now note that
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<cmath>m=\dfrac{27}{27\cdot 512}+\dfrac{3}{128}+\dfrac{1}{16}=\dfrac{1}{512}+\dfrac{12}{512}+\dfrac{32}{512}=\dfrac{45}{512}</cmath>
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and
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<cmath>n=\dfrac{-3}{24}-\dfrac{9}{576}=\dfrac{-9}{64}</cmath>
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Thus
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<cmath>r=\sqrt[3]{\dfrac{45}{512}+\sqrt{\dfrac{45^2}{512^2}-\dfrac{9^3}{64^3}}}+\sqrt[3]{\dfrac{45}{512}-\sqrt{\dfrac{45^2}{512^2}-\dfrac{9^3}{64^3}}}+\dfrac{3}{24}</cmath>
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<cmath>=\sqrt[3]{\dfrac{45}{512}+\sqrt{\dfrac{45^2 - 729}{2^{18}}}}+\sqrt[3]{\dfrac{45}{512}-\sqrt{\dfrac{45^2 - 729}{2^{18}}}}+\dfrac{1}{8}</cmath>
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<cmath>=\sqrt[3]{\dfrac{45}{512}+\dfrac{36}{512}}+\sqrt[3]{\dfrac{45}{512}-\dfrac{36}{512}}+\dfrac{1}{8}</cmath>
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<cmath>=\dfrac{\sqrt[3]{81}}{8}+\dfrac{\sqrt[3]{9}}{8}+\dfrac{1}{8}=\dfrac{\sqrt[3]{81}+\sqrt[3]{9}+1}{8}</cmath>
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== See Also ==
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{{AIME box|year=1983|before=First Question|num-a=2}}
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[[Category:Intermediate Algebra Problems]]

Revision as of 17:41, 16 March 2013

Problem

The real root of the equation $8x^3 - 3x^2 - 3x - 1 = 0$ can be written in the form $\frac{\sqrt[3]a + \sqrt[3]b + 1}{c}$, where $a$, $b$, and $c$ are positive integers. Find $a+b+c$.

Solutions

Solution 1

We have that $9x^3 = (x+1)^3$, so it follows that $\sqrt[3]{9}x = x+1$. Solving for $x$ yields $\frac{1}{\sqrt[3]{9}-1} = \frac{\sqrt[3]{81}+\sqrt[3]{9}+1}{8}$, so the answer is $\boxed{98}$.

Solution 2

Let $r$ be the real root of the given polynomial. Now define the cubic polynomial $Q(x)=-x^3-3x^2-3x+8$. Note that $1/r$ must be a root of $Q$. However we can simplify $Q$ as $Q(x)=9-(x+1)^3$, so we must have that $(\frac{1}{r}+1)^3=9$. Thus $\frac{1}{r}=\sqrt[3]{9}-1$, and $r=\frac{1}{\sqrt[3]{9}-1}$. We can then multiply the numerator and denominator of $r$ by $\sqrt[3]{81}+\sqrt[3]{9}+1$ to rationalize the denominator, and we therefore have $r=\frac{\sqrt[3]{81}+\sqrt[3]{9}+1}{8}$, and the answer is $\boxed{98}$.

Solution 3

It is clear that for the algebraic degree of $x$ to be $3$ that there exists some cubefree integer $p$ and positive integers $m,n$ such that $a = m^3p$ and $b = n^3p^2$ (it is possible that $b = n^3p$, but then the problem wouldn't ask for both an $a$ and $b$). Let $f_1$ be the automorpism over $\mathbb{Q}[\sqrt[3]{a}][\omega]$ which sends $\sqrt[3]{a} \to \omega \sqrt[3]{a}$ and $f_2$ which sends $\sqrt[3]{a} \to \omega^2 \sqrt[3]{a}$ (note : $\omega$ is a cubic root of unity).

Letting $r$ be the root, we clearly we have $r + f_1(r) + f_2(r) = \frac{3}{8}$ by Vieta's. Thus it follows $c=8$. Now, note that $\sqrt[3]{a} + \sqrt[3]{b} + 1$ is a root of $x^3 - 3x^2 - 24x - 64 = 0$. Thus $(x-1)^3 = 27x + 63$ so $(\sqrt[3]{a} + \sqrt[3]{b})^3 = 27(\sqrt[3]{a} + \sqrt[3]{b}) + 90$. Checking the non-cubicroot dimension part, we get $a + b = 90$ so it follows that $a + b + c = \boxed{98}$.

Solution 4

We proceed by using the cubic formula.

Let $a=8$, $b=-3$, $c=-3$, and $d=-1$. Then let $m=\left(\dfrac{-b^3}{27a^3}+\dfrac{bc}{6a^2}-\dfrac{d}{2a}\right)$ and $n=\left(\dfrac{c}{3a}-\dfrac{b^2}{9a^2}\right)$. Then the real root of $ax^3+bx^2+cx+d$ is \[\sqrt[3]{m+\sqrt{m^2+n^3}}+\sqrt[3]{m-\sqrt{m^2+n^3}}-\dfrac{b}{3a}\] Now note that \[m=\dfrac{27}{27\cdot 512}+\dfrac{3}{128}+\dfrac{1}{16}=\dfrac{1}{512}+\dfrac{12}{512}+\dfrac{32}{512}=\dfrac{45}{512}\] and \[n=\dfrac{-3}{24}-\dfrac{9}{576}=\dfrac{-9}{64}\] Thus \[r=\sqrt[3]{\dfrac{45}{512}+\sqrt{\dfrac{45^2}{512^2}-\dfrac{9^3}{64^3}}}+\sqrt[3]{\dfrac{45}{512}-\sqrt{\dfrac{45^2}{512^2}-\dfrac{9^3}{64^3}}}+\dfrac{3}{24}\] \[=\sqrt[3]{\dfrac{45}{512}+\sqrt{\dfrac{45^2 - 729}{2^{18}}}}+\sqrt[3]{\dfrac{45}{512}-\sqrt{\dfrac{45^2 - 729}{2^{18}}}}+\dfrac{1}{8}\] \[=\sqrt[3]{\dfrac{45}{512}+\dfrac{36}{512}}+\sqrt[3]{\dfrac{45}{512}-\dfrac{36}{512}}+\dfrac{1}{8}\] \[=\dfrac{\sqrt[3]{81}}{8}+\dfrac{\sqrt[3]{9}}{8}+\dfrac{1}{8}=\dfrac{\sqrt[3]{81}+\sqrt[3]{9}+1}{8}\]

See Also

1983 AIME (Problems • Answer Key • Resources)
Preceded by
First Question 		Followed by
Problem 2
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions
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