Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2021 AMC 10B Problems/Problem 15"

(Created page with "just go back and look at last year's")
 
(→‎Solution 7)
 
(32 intermediate revisions by 20 users not shown)
Line 1: Line 1:
−
just go back and look at last year's
+
==Problem==
 +
 
 +
The real number <math>x</math> satisfies the equation <math>x+\frac{1}{x} = \sqrt{5}</math>. What is the value of <math>x^{11}-7x^{7}+x^3?</math>
 +
 
 +
<math>\textbf{(A)} ~-1 \qquad\textbf{(B)} ~0 \qquad\textbf{(C)} ~1 \qquad\textbf{(D)} ~2 \qquad\textbf{(E)} ~\sqrt{5}</math>
 +
 
 +
==Solution 1==
 +
 
 +
We square <math>x+\frac{1}{x}=\sqrt5</math> to get <math>x^2+2+\frac{1}{x^2}=5</math>. We subtract 2 on both sides for <math>x^2+\frac{1}{x^2}=3</math> and square again, and see that <math>x^4+2+\frac{1}{x^4}=9</math> so <math>x^4+\frac{1}{x^4}=7</math>. We can factor out <math>x^7</math> from our original expression of <math>x^{11}-7x^7+x^3</math> to get that it is equal to <math>x^7(x^4-7+\frac{1}{x^4})</math>. Therefore because <math>x^4+\frac{1}{x^4}</math> is 7, it is equal to <math>x^7(0)=\boxed{\textbf{(B) } 0}</math>.
 +
 
 +
==Solution 2==
 +
 
 +
Multiplying both sides by <math>x</math> and using the quadratic formula, we get <math>\frac{\sqrt{5} \pm 1}{2}</math>. We can assume that it is <math>\frac{\sqrt{5}+1}{2}</math>, and notice that this is also a solution the equation <math>x^2-x-1=0</math>, i.e. we have <math>x^2=x+1</math>. Repeatedly using this on the given (you can also just note Fibonacci numbers),
 +
<cmath> \begin{align*}
 +
(x^{11})-7x^7+x^3 &= (x^{10}+x^9)-7x^7+x^3 \\
 +
&=(2x^9+x^8)-7x^7+x^3 \\
 +
&=(3x^8+2x^7)-7x^7+x^3 \\
 +
&=(3x^8-5x^7)+x^3 \\
 +
&=(-2x^7+3x^6)+x^3 \\
 +
&=(x^6-2x^5)+x^3 \\
 +
&=(-x^5+x^4+x^3) \\
 +
&=-x^3(x^2-x-1) = \boxed{\textbf{(B) } 0}
 +
\end{align*}</cmath>
 +
 
 +
~Lcz
 +
 
 +
==Solution 3==
 +
We can immediately note that the exponents of <math>x^{11}-7x^7+x^3</math> are an arithmetic sequence, so they are symmetric around the middle term. So, <math>x^{11}-7x^7+x^3 = x^7(x^4-7+\frac{1}{x^4})</math>. We can see that since <math>x+\frac{1}{x} = \sqrt{5}</math>, <math>x^2+2+\frac{1}{x^2} = 5</math> and therefore <math>x^2+\frac{1}{x^2} = 3</math>. Continuing from here, we get <math>x^4+2+\frac{1}{x^4} = 9</math>, so <math>x^4-7+\frac{1}{x^4} = 0</math>. We don't even need to find what <math>x^7</math> is! This is since <math>x^7\cdot0</math> is evidently <math>\boxed{\textbf{(B) } 0}</math>, which is our answer.
 +
 
 +
~sosiaops
 +
 
 +
==Solution 4==
 +
We begin by multiplying <math>x+\frac{1}{x} = \sqrt{5}</math> by <math>x</math>, resulting in <math>x^2+1 = \sqrt{5}x</math>. Now we see this equation: <math>x^{11}-7x^{7}+x^3</math>. The terms all have <math>x^3</math> in common, so we can factor that out, and what we're looking for becomes <math>x^3(x^8-7x^4+1)</math>. Looking back to our original equation, we have <math>x^2+1 = \sqrt{5}x</math>, which is equal to <math>x^2 = \sqrt{5}x-1</math>. Using this, we can evaluate <math>x^4</math> to be <math>5x^2-2\sqrt{5}x+1</math>, and we see that there is another <math>x^2</math>, so we put substitute it in again, resulting in <math>3\sqrt{5}x-4</math>. Using the same way, we find that <math>x^8</math> is <math>21\sqrt{5}x-29</math>. We put this into <math>x^3(x^8-7x^4+1)</math>, resulting in <math>x^3(0)</math>, so the answer is <math>\boxed{(B)~0}</math>.
 +
 
 +
~purplepenguin2
 +
 
 +
==Solution 5==
 +
The equation we are given is <math>x+\tfrac{1}{x}=\sqrt{5}...</math> Yuck. Fractions and radicals! We multiply both sides by <math>x,</math> square, and re-arrange to get <cmath>x^2+1=\sqrt{5}x \implies x^4+2x^2+1=5x^2 \implies x^4-3x^2+1=0.</cmath> Now, let us consider the expression we wish to acquire. Factoring out <math>x^3,</math> we have <cmath>x^3\left(x^8-7x^4+1\right) = x^3\left(x^8+2x^4+1-9x^4\right).</cmath> Then, we notice that <math>x^8+2x^4+1=\left(x^4+1\right)^2.</math> Furthermore, <cmath>x^4+1=3x^2 \implies \left(x^4+1\right)^2=x^8+2x^4+1=9x^4.</cmath> Thus, our answer is <cmath>x^3\left(9x^4-9x^4\right) = x^3 \cdot 0 = \boxed{\textbf{(B)}} ~ 0.</cmath>
 +
~peace09
 +
==Solution 6(Non-rigorous for little time)==
 +
Multiplying by x and solving, we get that <math>x = \frac{\sqrt{5} \pm 1}{2}.</math> Note that whether or not we take <math>x = \frac{\sqrt{5} + 1}{2}</math> or we take <math>\frac{\sqrt{5} - 1}{2},</math> our answer has to be the same. Thus, we take <math>x = \frac{\sqrt{5} - 1}{2} \approx 0.62</math>. Since this number is small, taking it to high powers like <math>11</math>, <math>7</math>, and <math>3</math> will make the number very close to <math>0</math>, so the answer is <math>\boxed{(B)~0}.</math>
 +
~AtharvNaphade
 +
 
 +
==Solution 7==
 +
We know that <math>x+\frac{1}{x}=\sqrt{5}</math>. Multiply both sides by <math>x</math> to get <math>x^2+1=x\sqrt{5}</math>
 +
Squaring both sides: <cmath>x^4+2x^2+1=5x^2</cmath>
 +
Subtract <math>2x^2</math> from both sides: <cmath>x^4+1=3x^2</cmath>
 +
Squaring both sides: <cmath>x^8+2x^4+1=9x^4</cmath>
 +
Subtract <math>9x^4</math> from both sides: <cmath>x^8-7x^4+1=0</cmath>
 +
Multiply <math>x^3</math> on both sides: <cmath>x^{11}-7x^7+x^3=\fbox{(B) 0}</cmath>
 +
~sid2012 [https://artofproblemsolving.com/wiki/index.php/User:Sid2012]
 +
 
 +
==Video Solution (🚀 Super Fast. Under 2 min! 🚀)==
 +
https://youtu.be/CJbtpNhMvIM
 +
 
 +
<i> ~Education, the Study of Everything </i>
 +
 
 +
== Video Solution by OmegaLearn ==
 +
https://youtu.be/M4Ffhp9NLKY?t=81
 +
 
 +
~ pi_is_3.14
 +
 
 +
== Video Solution by Interstigation (Simple Silly Bashing) ==
 +
https://youtu.be/Hdk2SDOcw7c
 +
 
 +
~ Interstigation
 +
 
 +
==Video Solution by TheBeautyofMath==
 +
Not the most efficient method, but gets the job done.
 +
 
 +
https://youtu.be/L1iW94Ue3eI?t=1468
 +
 
 +
~IceMatrix
 +
 
 +
==See Also==
 +
{{AMC10 box|year=2021|ab=B|num-b=14|num-a=16}}
 +
{{MAA Notice}}

Latest revision as of 13:06, 29 September 2024

Problem

The real number $x$ satisfies the equation $x+\frac{1}{x} = \sqrt{5}$. What is the value of $x^{11}-7x^{7}+x^3?$

$\textbf{(A)} ~-1 \qquad\textbf{(B)} ~0 \qquad\textbf{(C)} ~1 \qquad\textbf{(D)} ~2 \qquad\textbf{(E)} ~\sqrt{5}$

Solution 1

We square $x+\frac{1}{x}=\sqrt5$ to get $x^2+2+\frac{1}{x^2}=5$. We subtract 2 on both sides for $x^2+\frac{1}{x^2}=3$ and square again, and see that $x^4+2+\frac{1}{x^4}=9$ so $x^4+\frac{1}{x^4}=7$. We can factor out $x^7$ from our original expression of $x^{11}-7x^7+x^3$ to get that it is equal to $x^7(x^4-7+\frac{1}{x^4})$. Therefore because $x^4+\frac{1}{x^4}$ is 7, it is equal to $x^7(0)=\boxed{\textbf{(B) } 0}$.

Solution 2

Multiplying both sides by $x$ and using the quadratic formula, we get $\frac{\sqrt{5} \pm 1}{2}$. We can assume that it is $\frac{\sqrt{5}+1}{2}$, and notice that this is also a solution the equation $x^2-x-1=0$, i.e. we have $x^2=x+1$. Repeatedly using this on the given (you can also just note Fibonacci numbers), \begin{align*} (x^{11})-7x^7+x^3 &= (x^{10}+x^9)-7x^7+x^3 \\ &=(2x^9+x^8)-7x^7+x^3 \\ &=(3x^8+2x^7)-7x^7+x^3 \\ &=(3x^8-5x^7)+x^3 \\ &=(-2x^7+3x^6)+x^3 \\ &=(x^6-2x^5)+x^3 \\ &=(-x^5+x^4+x^3) \\ &=-x^3(x^2-x-1) = \boxed{\textbf{(B) } 0} \end{align*}

~Lcz

Solution 3

We can immediately note that the exponents of $x^{11}-7x^7+x^3$ are an arithmetic sequence, so they are symmetric around the middle term. So, $x^{11}-7x^7+x^3 = x^7(x^4-7+\frac{1}{x^4})$. We can see that since $x+\frac{1}{x} = \sqrt{5}$, $x^2+2+\frac{1}{x^2} = 5$ and therefore $x^2+\frac{1}{x^2} = 3$. Continuing from here, we get $x^4+2+\frac{1}{x^4} = 9$, so $x^4-7+\frac{1}{x^4} = 0$. We don't even need to find what $x^7$ is! This is since $x^7\cdot0$ is evidently $\boxed{\textbf{(B) } 0}$, which is our answer.

~sosiaops

Solution 4

We begin by multiplying $x+\frac{1}{x} = \sqrt{5}$ by $x$, resulting in $x^2+1 = \sqrt{5}x$. Now we see this equation: $x^{11}-7x^{7}+x^3$. The terms all have $x^3$ in common, so we can factor that out, and what we're looking for becomes $x^3(x^8-7x^4+1)$. Looking back to our original equation, we have $x^2+1 = \sqrt{5}x$, which is equal to $x^2 = \sqrt{5}x-1$. Using this, we can evaluate $x^4$ to be $5x^2-2\sqrt{5}x+1$, and we see that there is another $x^2$, so we put substitute it in again, resulting in $3\sqrt{5}x-4$. Using the same way, we find that $x^8$ is $21\sqrt{5}x-29$. We put this into $x^3(x^8-7x^4+1)$, resulting in $x^3(0)$, so the answer is $\boxed{(B)~0}$.

~purplepenguin2

Solution 5

The equation we are given is $x+\tfrac{1}{x}=\sqrt{5}...$ Yuck. Fractions and radicals! We multiply both sides by $x,$ square, and re-arrange to get \[x^2+1=\sqrt{5}x \implies x^4+2x^2+1=5x^2 \implies x^4-3x^2+1=0.\] Now, let us consider the expression we wish to acquire. Factoring out $x^3,$ we have \[x^3\left(x^8-7x^4+1\right) = x^3\left(x^8+2x^4+1-9x^4\right).\] Then, we notice that $x^8+2x^4+1=\left(x^4+1\right)^2.$ Furthermore, \[x^4+1=3x^2 \implies \left(x^4+1\right)^2=x^8+2x^4+1=9x^4.\] Thus, our answer is \[x^3\left(9x^4-9x^4\right) = x^3 \cdot 0 = \boxed{\textbf{(B)}} ~ 0.\] ~peace09

Solution 6(Non-rigorous for little time)

Multiplying by x and solving, we get that $x = \frac{\sqrt{5} \pm 1}{2}.$ Note that whether or not we take $x = \frac{\sqrt{5} + 1}{2}$ or we take $\frac{\sqrt{5} - 1}{2},$ our answer has to be the same. Thus, we take $x = \frac{\sqrt{5} - 1}{2} \approx 0.62$. Since this number is small, taking it to high powers like $11$, $7$, and $3$ will make the number very close to $0$, so the answer is $\boxed{(B)~0}.$ ~AtharvNaphade

Solution 7

We know that $x+\frac{1}{x}=\sqrt{5}$. Multiply both sides by $x$ to get $x^2+1=x\sqrt{5}$ Squaring both sides: \[x^4+2x^2+1=5x^2\] Subtract $2x^2$ from both sides: \[x^4+1=3x^2\] Squaring both sides: \[x^8+2x^4+1=9x^4\] Subtract $9x^4$ from both sides: \[x^8-7x^4+1=0\] Multiply $x^3$ on both sides: \[x^{11}-7x^7+x^3=\fbox{(B) 0}\] ~sid2012 [1]

Video Solution (🚀 Super Fast. Under 2 min! 🚀)

https://youtu.be/CJbtpNhMvIM

~Education, the Study of Everything

Video Solution by OmegaLearn

https://youtu.be/M4Ffhp9NLKY?t=81

~ pi_is_3.14

Video Solution by Interstigation (Simple Silly Bashing)

https://youtu.be/Hdk2SDOcw7c

~ Interstigation

Video Solution by TheBeautyofMath

Not the most efficient method, but gets the job done.

https://youtu.be/L1iW94Ue3eI?t=1468

~IceMatrix

See Also

2021 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 14 		Followed by
Problem 16
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 10 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2021_AMC_10B_Problems/Problem_15&oldid=228611"